Introduction
Interior point methods (IPMs) are among the most powerful algorithmic tools for continuous optimization. Their success is closely tied to convexity, which provides the structure needed to control Newton steps and guarantee rapid convergence. A longstanding question is how far these methods can be extended beyond convex optimization, and what properties can replace convexity in their convergence analysis.
An important setting for studying this question is the linear complementarity problem (LCP). Given a matrix M ∈ R n × n M \in\mathbb{R}^{n \times n} M ∈ R n × n q ∈ R n q \in\mathbb{R}^n q ∈ R n x , s ∈ R n x,s \in\mathbb{R}^n x , s ∈ R n
M x + q = s , x T s = 0 , x , s ≥ 0 n . (1) Mx + q = s,\qquad x^\mathsf{T}s = 0,\qquad x,s \geq0_n. \tag*{(1)} M x + q = s , x T s = 0 , x , s ≥ 0 n . (1) The LCP encompasses a remarkably broad range of problems. In particular, the optimality conditions of linear and convex quadratic programs can be expressed as LCPs, and complementarity formulations arise naturally in economic equilibria, noncooperative games, and variational inequalities. They were first studied in the 1960s by Lemke [41 ] and by Cottle and Dantzig [3 ] , extending the complementary pivot algorithm by Lemke and Howson [42 ] from bimatrix games to the more general setting. We refer the reader to the books by Cottle, Pang, and Stone [8 ] and by Murty and Yu [52 ] for comprehensive treatments of LCP.
The best understood case is that of monotone LCPs, where M + M T M + M^\mathsf{T} M + M T x T M x ≥ 0 x^\mathsf{T}Mx \geq0 x T M x ≥ 0 x ∈ R n x \in\mathbb{R}^n x ∈ R n [39 , 15 , 40 , 86 , 85 ] . Unfortunately, many important LCPs do not enjoy this structure. For unrestricted matrices, solving an LCP is NP-hard [5 ] , and the complexity of several natural matrix classes remains a major open problem.
A particularly important example is the class of P \mathcal{P} P P \mathcal{P} P M M M ( M , q ) (M,q) ( M , q ) q q q [73 ] . Cottle and Dantzig [3 ] showed that Lemke’s algorithm finitely finds a solution for this class, yet there are still no polynomial time algorithms known. Megiddo [49 ] showed that solving P P P P P P [74 , 75 ] and to the complexity of total search problems: they are among the first problems shown to lie in PPAD by Papadimitriou [55 ] , and also belong to the subclass UEOPL of PPAD [20 ] . Prominent problems at the intersection of automata theory and algorithmic game theory, such as discounted stochastic games, mean-payoff and parity games, admit reductions to P P P [25 , 34 , 29 , 20 ] . These connections make the search for efficient algorithms for structured LCPs an important direction in theoretical computer science and in optimization.
Analogously to their distinguished role in the context of pivoting algorithms, P P P [38 ] introduced the parametrized matrix class P ⋆ ( κ ) P_{\star}(\kappa) P ⋆ ( κ ) P ⋆ ( κ ) P_{\star}(\kappa) P ⋆ ( κ ) κ \kappa κ κ \kappa κ handicap number , measures the extent to which a matrix violates monotonicity; every P P P P ⋆ ( κ ) P_{\star}(\kappa) P ⋆ ( κ ) κ ≥ 0 \kappa\ge0 κ ≥ 0 P P P κ \kappa κ
This was followed by a long line of work on developing IPMs for P ⋆ ( κ ) P_{\star}(\kappa) P ⋆ ( κ ) [50 , 62 , 63 , 60 , 56 , 43 , 64 , 61 , 44 , 33 , 18 , 11 ] , see also § 1.2 . The literature explored a variety of IPMs using different neighborhoods, short and long-steps, as well as different kernel functions and reparametrizations. However, the best known running time bounds for P ⋆ ( κ ) P_{\star}(\kappa) P ⋆ ( κ ) κ \kappa κ
In this paper, we make the first improvement to this dependence in over three decades, showing that it can be reduced from linear to square root. Among extensive efforts to improve IPM iteration complexity across diverse regimes [47 , 48 CMSV17, 45 , 2 , 37 , 83 , 46 ] , to the extent of our knowledge this is the first result to break a longstanding bound by reanalyzing the classical method in a norm dictated by combinatorial structure.
Handicap number and LCPs for P ⋆ P_{\star} P ⋆ To understand the handicap number, recall that monotonicity requires
x T M x = ∑ i = 1 n x i ( M x ) i ≥ 0 , for all x ∈ R n . x^{\mathsf{T}}Mx = \sum_{i=1}^{n} x_i(Mx)_i \ge0,\qquad\text{for all } x \in\mathbb{R}^{n}. x T M x = i = 1 ∑ n x i ( M x ) i ≥ 0 , for all x ∈ R n . Although the individual terms in this sum may be negative, their total must be nonnegative. The defining property of P ⋆ ( κ ) P_{\star}(\kappa) P ⋆ ( κ ) 1 + 4 κ 1+4\kappa 1 + 4 κ x ∈ R n x \in\mathbb{R}^{n} x ∈ R n x i ( M x ) i x_i(Mx)_i x i ( M x ) i
I M + ( x ) : = { i ∈ [ n ] : x i ( M x ) i > 0 } , and I M − ( x ) : = { i ∈ [ n ] : x i ( M x ) i < 0 } . \mathcal{I}^{+}_{M}(x) := \{i \in[n] : x_i(Mx)_i > 0\}, \qquad\text{and}\qquad\mathcal{I}^{-}_{M}(x) := \{i \in[n] : x_i(Mx)_i < 0\}. I M + ( x ) := { i ∈ [ n ] : x i ( M x ) i > 0 } , and I M − ( x ) := { i ∈ [ n ] : x i ( M x ) i < 0 } . The handicap number of M M M
κ ^ ( M ) : = inf { κ ≥ 0 : ( 1 + 4 κ ) ∑ i ∈ I M + ( x ) x i ( M x ) i + ∑ i ∈ I M − ( x ) x i ( M x ) i ≥ 0 , ∀ x ∈ R n } . (2) \widehat{\kappa}(M) := \inf\left\{ \kappa\ge0 : (1+4\kappa)\sum_{i\in\mathcal{I}^{+}_{M}(x)} x_i(Mx)_i + \sum_{i\in\mathcal{I}^{-}_{M}(x)} x_i(Mx)_i \ge0,\quad\forall x\in\mathbb{R}^{n} \right\}. \tag*{(2)} κ ( M ) := inf ⎩ ⎨ ⎧ κ ≥ 0 : ( 1 + 4 κ ) i ∈ I M + ( x ) ∑ x i ( M x ) i + i ∈ I M − ( x ) ∑ x i ( M x ) i ≥ 0 , ∀ x ∈ R n ⎭ ⎬ ⎫ . (2) We let P ⋆ ( κ ) P_{\star}(\kappa) P ⋆ ( κ ) κ ^ ( M ) ≤ κ \widehat{\kappa}(M) \le\kappa κ ( M ) ≤ κ P ⋆ : = ⋃ κ ≥ 0 P ⋆ ( κ ) P_{\star} := \bigcup_{\kappa\ge0} P_{\star}(\kappa) P ⋆ := ⋃ κ ≥ 0 P ⋆ ( κ ) P ⋆ ( 0 ) P_{\star}(0) P ⋆ ( 0 )
The class P ⋆ P_{\star} P ⋆ [81 ] (Theorem 2.3) shows that it coincides with the class of sufficient matrices , originally introduced by Cottle, Pang, and Venkateswaran [9 ] . A matrix M ∈ R n × n M \in\mathbb{R}^{n\times n} M ∈ R n × n column sufficient if for all x ∈ R n x \in\mathbb{R}^{n} x ∈ R n x ( M x ) ≤ 0 n x(Mx) \le0_n x ( M x ) ≤ 0 n x ( M x ) = 0 n x(Mx)=0_n x ( M x ) = 0 n row sufficient if M T M^{\mathsf{T}} M T x x x M x Mx M x sufficient if it is both column and row sufficient.
Moreover, P ⋆ P_{\star} P ⋆ P 0 P_0 P 0 P P P P 0 P_0 P 0 [38 Section 3.4] . Sufficient matrices fall strictly between these classes: P ⊊ P ⋆ ⊊ P 0 P \subsetneq P_{\star} \subsetneq P_0 P ⊊ P ⋆ ⊊ P 0 [38 Section 3.2] .
While the linear dependence on κ \kappa κ P \mathcal{P} P [29 ] showed that, for games with n n n γ ∈ ( 0 , 1 ) \gamma\in(0,1) γ ∈ ( 0 , 1 ) O ( n ( 1 − γ ) 2 ) O\left(\frac{n}{(1-\gamma)^2}\right) O ( ( 1 − γ ) 2 n ) 1 / ( 1 − γ ) 1/(1-\gamma) 1/ ( 1 − γ ) 1 / ( 1 − γ ) 1/(1-\gamma) 1/ ( 1 − γ ) [31 ] ; see further discussion in § 1.2 .
This gap suggests that the handicap-based IPM analysis may be losing substantially more than necessary. The 1 + κ \sqrt{1+\kappa} 1 + κ P ⋆ ( κ ) \mathcal{P}_{\star}(\kappa) P ⋆ ( κ ) log ( 1 + κ ) \log(1+\kappa) log ( 1 + κ ) P ⋆ \mathcal{P}_{\star} P ⋆
Our contributions
New characterization of P ⋆ ( κ ) \mathcal{P}_{\star}(\kappa) P ⋆ ( κ ) Classical analyses of interior point methods for P ⋆ ( κ ) \mathcal{P}_{\star}(\kappa) P ⋆ ( κ ) κ ^ ( M ) \hat{\kappa}(M) κ ^ ( M ) A ∈ R m × n A \in\mathbb{R}^{m \times n} A ∈ R m × n ∥ A ∥ 2 = ∥ A ∥ 2 → 2 = max x ∈ R n , ∥ x ∥ 2 = 1 ∥ A x ∥ \|A\|_2 = \|A\|_{2 \to2} = \max_{x \in\mathbb{R}^n,\|x\|_2=1} \|Ax\| ∥ A ∥ 2 = ∥ A ∥ 2 → 2 = max x ∈ R n , ∥ x ∥ 2 = 1 ∥ A x ∥ ℓ 2 → ℓ 2 \ell_2 \to\ell_2 ℓ 2 → ℓ 2 D > 0 n ⊆ R n × n \mathcal{D}_{>0}^n \subseteq\mathbb{R}^{n \times n} D > 0 n ⊆ R n × n
Theorem 1.1. Let M ∈ R n × n M \in\mathbb{R}^{n \times n} M ∈ R n × n M ∈ P ⋆ M \in\mathcal{P}_{\star} M ∈ P ⋆ I n + D M D I_n + DMD I n + D M D D ∈ D > 0 n D \in\mathcal{D}_{>0}^n D ∈ D > 0 n sup D ∈ D > 0 n ∥ ( I n + D M D ) − 1 ∥ 2 \sup_{D \in\mathcal{D}_{>0}^n}\|(I_n + DMD)^{-1}\|_2 sup D ∈ D > 0 n ∥ ( I n + D M D ) − 1 ∥ 2 M ∈ P ⋆ M \in\mathcal{P}_{\star} M ∈ P ⋆
sup D ∈ D > 0 n ∥ ( I n + D M D ) − 1 ∥ 2 = 1 + κ ^ ( M ) 2 1 / 4 + κ ^ ( M ) . (3) \sup_{D \in\mathcal{D}_{>0}^n}\|(I_n + DMD)^{-1}\|_2 = \sqrt{1 + \frac{\hat{\kappa}(M)^2}{1/4 + \hat{\kappa}(M)}}. \tag*{(3)} D ∈ D > 0 n sup ∥ ( I n + D M D ) − 1 ∥ 2 = 1 + 1/4 + κ ^ ( M ) κ ^ ( M ) 2 . (3) The proof is given in § 9.1 . Besides being the key to our improved analysis, it also leads to simple new proofs of a theorem of Väliaho [82 ] showing that the handicap number is invariant under transposition, see Theorem 2.6.
Improved IPM analysis via algebraic and combinatorial properties. As we will see in Section 4, Newton steps for interior point methods involve matrix inverses of the form as in (2). In Theorem 4.2 , we give a simple analysis showing that O ( n ( 1 + κ ^ ( M ) ) log ( ρ ) ) O(\sqrt{n}(1+\hat{\kappa}(M))\log(\rho)) O ( n ( 1 + κ ^ ( M )) log ( ρ )) ρ \rho ρ 1 + κ ^ ( M ) \sqrt{1+\hat{\kappa}(M)} 1 + κ ^ ( M )
Theorem 1.2. Let M ∈ P ⋆ ∩ Q n × n M \in\mathcal{P}_{\star} \cap\mathbb{Q}^{n \times n} M ∈ P ⋆ ∩ Q n × n q ∈ Q n q \in\mathbb{Q}^n q ∈ Q n L L L LCP ( M , q ) \operatorname{LCP}(M,q) LCP ( M , q )
O ( min { n ( 1 + κ ^ ( M ) ) , n 9 / 8 1 + κ ^ ( M ) } ⋅ ( L + log ( 2 + κ ^ ( M ) ) ) ) O\left(\min\left\{\sqrt{n}(1+\hat{\kappa}(M)), n^{9/8}\sqrt{1+\hat{\kappa}(M)}\right\}\cdot\left(L+\log(2+\hat{\kappa}(M))\right)\right) O ( min { n ( 1 + κ ^ ( M )) , n 9/8 1 + κ ^ ( M ) } ⋅ ( L + log ( 2 + κ ^ ( M )) ) ) many linear system solves, and in addition the running time of O ( log ( 2 + κ ^ ( M ) ) ) O(\log(2+\hat{\kappa}(M))) O ( log ( 2 + κ ^ ( M ))) M x + q = s Mx+q=s M x + q = s x , s ≥ 0 n x,s \geq0_n x , s ≥ 0 n x = 0 I x=0_I x = 0 I s = 0 J s=0_J s = 0 J I , J ⊆ [ n ] I,J \subseteq[n] I , J ⊆ [ n ]
In contrast with the more sophisticated IPM algorithms developed over the past decades, our result uses standard Newton steps, for which we provide a significantly stronger analysis. Analyzing convergence in the Euclidean norm naturally leads to linear κ ^ ( M ) + 1 \hat{\kappa}(M)+1 κ ^ ( M ) + 1 κ ^ ( M ) \hat{\kappa}(M) κ ^ ( M ) O ( 1 + κ ^ ( M ) ) O(\sqrt{1+\hat{\kappa}(M)}) O ( 1 + κ ^ ( M ) )
The construction relies on a remarkable theorem of Rump [66 , 67 ] , relating the sign-real spectral radius of a matrix to the maximum geometric mean of products of entries of the matrix along directed cycles. While Rump established a strong connection between the sign-real spectral radius and P \mathcal{P} P P ⋆ \mathcal{P}_{\star} P ⋆
New hardness results. It is natural to ask for the complexity status of recognizing P ⋆ \mathcal{P}_\star P ⋆ P \mathcal{P} P P 0 \mathcal{P}_0 P 0 [77 ] . However, these do not imply the complexity status of recognizing sufficient matrices, which remains an open problem despite the wide belief that it is coNP-complete.1 P ~ \widetilde{\mathcal{P}} P P \mathcal{P} P P 0 \mathcal{P}_0 P 0
Theorem 1.3. For any matrix class P ~ \widetilde{\mathcal{P}} P P ⊆ P ~ ⊆ P 0 \mathcal{P} \subseteq\widetilde{\mathcal{P}} \subseteq\mathcal{P}_0 P ⊆ P ⊆ P 0 M ∈ Q n × n M \in\mathbb{Q}^{n \times n} M ∈ Q n × n P ~ \widetilde{\mathcal{P}} P M ∈ Q n × n M \in\mathbb{Q}^{n \times n} M ∈ Q n × n P ⋆ \mathcal{P}_\star P ⋆
Interestingly, the proof also relies heavily on the notion of sign-real spectral radius. For any M ∈ Q n × n M \in\mathbb{Q}^{n \times n} M ∈ Q n × n M ~ ∈ Q n × n \widetilde{M} \in\mathbb{Q}^{n \times n} M ∈ Q n × n M ~ ∈ P ~ \widetilde{M} \in\widetilde{\mathcal{P}} M ∈ P M ∈ P M \in\mathcal{P} M ∈ P P \mathcal{P} P Theorem 7.4 ). Then the result follows as a consequence of coNP-completeness of recognizing P \mathcal{P} P
Given a promise that a matrix is sufficient, computing its handicap number is still considered to be difficult. As shown in [14 , 17 ] , it can grow exponentially with the matrix dimension. It was only recently proven that the logarithm of the handicap number can be polynomially bounded in the encoding length [19 ] . The best known algorithm to compute handicap number remains Väliaho’s recursive algorithm [82 ] through enumeration over principal submatrices of principal pivot transforms, which is exponential.
Our next result shows approximation hardness of computing the handicap number of an upper triangular matrix. Recall that in the Max-Cut problem, the input is an undirected graph G = ( V , E ) G = (V, E) G = ( V , E ) S ⊆ V S \subseteq V S ⊆ V S S S V ∖ S V \setminus S V ∖ S
Theorem 1.4. A polynomial-time multiplicative α \alpha α 1 + 4 κ ^ ( M ) 1 + 4\widehat{\kappa}(M) 1 + 4 κ ( M ) M ∈ P ∩ Q n × n M \in\mathcal{P} \cap\mathbb{Q}^{n \times n} M ∈ P ∩ Q n × n α \alpha α Max-Cut .
As shown in [36 ] , Max-Cut cannot be approximated by a factor better than the reciprocal of the approximation ratio ≈ 0.878 \approx0.878 ≈ 0.878 [26 ] , assuming the Unique Games Conjecture.2 17 / 16 17/16 17/16 [28 , 78 ] . We prove our result by giving a construction of an upper triangular P \mathcal{P} P M G M_G M G G G G 1 + 4\widehat{\kappa}(M_G) = 4\maxcut(G) . Our proof is motivated by the connection between the sign-real spectral radius and Max-Cut. Rump showed [66 ] that the quantity max s 1 , s 2 ∈ { − 1 , 1 } n s 1 T M s 2 \max_{s_1,s_2 \in\{-1,1\}^n} s_1^{\mathsf T}Ms_2 max s 1 , s 2 ∈ { − 1 , 1 } n s 1 T M s 2 M M M [59 ] that Max-Cut can be reduced to computing max s 1 , s 2 ∈ { − 1 , 1 } n s 1 T M s 2 \max_{s_1,s_2 \in\{-1,1\}^n} s_1^{\mathsf T}Ms_2 max s 1 , s 2 ∈ { − 1 , 1 } n s 1 T M s 2 M M M P ⋆ \mathcal{P}_\star P ⋆
Further related work
The first polynomial time IPM for solving monotone LCPs was given by Kojima, Mizuno and Yoshise [39 ] , followed by further algorithms including [15 , 40 , 86 , 85 ] . The book by Kojima, Megiddo, Noma, and Yoshise [38 ] discussed above extended these methods for P ⋆ \mathcal{P}_\star P ⋆ [50 ] established an O ~ ( ( 1 + κ ) n ) \widetilde{O}((1+\kappa)\sqrt{n}) O (( 1 + κ ) n ) [51 ] . Potra and Sheng [62 , 63 ] developed variants with infeasible starting point. Potra and Liu [56 , 43 ] introduced wide neighborhood algorithms which do not require the explicit handicap number as input. Peng, Roos, and Terlaky [60 , 61 ] developed IPMs that use more general barrier functions called self-regular kernel functions, and used these to derive long-step IPMs for P ⋆ \mathcal{P}_\star P ⋆ [44 ] extended the analysis for a broader class of eligible kernel functions. Darvay [10 ] introduced the algebraic equivalent transformation technique for LP, a certain reparametrization of the central path. This has been extended for P ⋆ \mathcal{P}_\star P ⋆ [18 , 13 ] . The recent paper [11 ] presents a unified analysis of such algorithms. P ⋆ \mathcal{P}_\star P ⋆ [57 , 58 ] , see also [16 ] . Despite considerable progress in algorithm design and convergence analysis, the best general handicap dependence in all these works remained linear.
Sufficient matrices were introduced in [9 ] . Characterizations were given by Cottle and Guu in [4 , 24 ] , in particular, the latter paper conjectured that this class equals P ⋆ \mathcal{P}_{\star} P ⋆ [81 ] , see also [82 ] and [80 ] for further properties and characterizations. The sign-real spectral radius of a matrix, which plays a key role in our work, was further studied by Rump [67 , 68 , 69 ] .
E.-Nagy and Végh [19 ] introduced the optimized handicap number κ ^ ⋆ ( M ) \hat{\kappa}^{\star}(M) κ ^ ⋆ ( M ) κ ^ ( D M ) \hat{\kappa}(DM) κ ^ ( D M ) LCP ( M , q ) \operatorname{LCP}(M,q) LCP ( M , q ) poly ( n , L , κ ^ ⋆ ( M ) ) \operatorname{poly}(n,L,\hat{\kappa}^{\star}(M)) poly ( n , L , κ ^ ⋆ ( M )) κ ^ ⋆ ( M ) \hat{\kappa}^{\star}(M) κ ^ ⋆ ( M ) 1 + κ ^ ⋆ ( M ) \sqrt{1+\hat{\kappa}^{\star}(M)} 1 + κ ^ ⋆ ( M )
In the context of Theorem 1.4 on approximation hardness of the handicap number, it is worth noting that for some other matrix condition measures relevant in optimization, significantly stronger inapproximability results are known. The smallest absolute value subdeterminant [35 ] , the Dikin–Stewart–Todd condition number χ ˉ M \bar{\chi}_{M} χ ˉ M [79 ] , and the circuit imbalance measure [12 ] cannot be approximated by a factor better than 2 poly ( n ) 2^{\operatorname{poly}(n)} 2 poly ( n ) 2 O ( n ) 2^{O(n)} 2 O ( n ) log ( 2 + κ ^ ( M ) ) \log(2+\hat{\kappa}(M)) log ( 2 + κ ^ ( M ))
Among the applications of P ⋆ \mathcal{P}_{\star} P ⋆ [72 ] . Zwick and Paterson also reduced mean payoff games to this problem [87 ] . As mentioned, it was formulated as a P \mathcal{P} P [34 ] . Hansen, Miltersen and Zwick [31 ] showed that Howard’s policy iteration algorithm terminates in O ( m 1 − γ log n 1 − γ ) O\left(\frac{m}{1-\gamma}\log\frac{n}{1-\gamma}\right) O ( 1 − γ m log 1 − γ n ) n n n m m m [27 ] gave the first randomized subexponential algorithm, with the current best e O ( n log ( m / n ) ) e^{O(\sqrt{n\log(m/n)})} e O ( n l o g ( m / n ) ) [88 ] . Very recently, quasipolynomial algorithms were announced by Soleymani and Farina [71 ] and by an AI company [53 ] . Using the handicap bound κ = O ( n / ( 1 − γ ) 2 ) \kappa=O(n/(1-\gamma)^2) κ = O ( n / ( 1 − γ ) 2 ) [29 ] , our algorithm has running time O ( n 1.625 L / ( 1 − γ ) ) O(n^{1.625}L/(1-\gamma)) O ( n 1.625 L / ( 1 − γ ))
Roadmap
The rest of this paper is organized as follows.
Section 2 – we cover preliminaries, such as notation, basic properties of P ⋆ ( κ ) \mathcal{P}_{\star}(\kappa) P ⋆ ( κ )
Section 5 – we introduce Rump weights, and use them to analyze the same algorithm in a changing weighted norm. We show that Newton’s method converges in this norm, improving the dependence on the handicap number from linear to square root.
Section 8 – we show how to initialize the IPM, recover an exact solution, and run the algorithm without prior knowledge of κ ^ ( M ) \hat{\kappa}(M) κ ^ ( M )
Preliminaries
Notation
Vectors and matrices. For n ∈ N n \in\mathbb{N} n ∈ N [ n ] : = { 1 , 2 , … , n } [n]:=\{1,2,\ldots,n\} [ n ] := { 1 , 2 , … , n } 0 n 0_n 0 n 1 n 1_n 1 n n n n I n I_n I n n × n n \times n n × n i ∈ [ n ] i \in[n] i ∈ [ n ] e i e_i e i i i i ( e i ) i = 1 (e_i)_i=1 ( e i ) i = 1 ( e i ) j = 0 (e_i)_j=0 ( e i ) j = 0 j ≠ i j \ne i j = i ⟨ x , y ⟩ \langle x,y\rangle ⟨ x , y ⟩ x T y x^{\mathsf{T}}y x T y x , y ∈ R n x,y \in\mathbb{R}^{n} x , y ∈ R n X : = diag ( x ) X:=\operatorname{diag}(x) X := diag ( x ) X i i = x i X_{ii}=x_i X ii = x i x x x n × n n \times n n × n D > 0 n D_{>0}^n D > 0 n D ≥ 0 n D_{\geq0}^n D ≥ 0 n A ∈ R n × n A \in\mathbb{R}^{n\times n} A ∈ R n × n A ⪰ 0 n A \succeq0_n A ⪰ 0 n x ⊤ A x ≥ 0 x^\top A x \geq0 x ⊤ A x ≥ 0 x ∈ R n x \in\mathbb{R}^n x ∈ R n A ∈ Q m × n A \in\mathbb{Q}^{m\times n} A ∈ Q m × n length ( A ) \operatorname{length}(A) length ( A ) A A A i ∈ [ n ] i \in[n] i ∈ [ n ] A ∈ R n × n A \in\mathbb{R}^{n\times n} A ∈ R n × n A ∙ i ∈ R n A_{\bullet i} \in\mathbb{R}^n A ∙ i ∈ R n i i i A A A A i ∙ A_{i\bullet} A i ∙ i i i
For x , y ∈ R n x,y \in\mathbb{R}^n x , y ∈ R n x y = diag ( x ) y = ( x i y i ) i ∈ [ n ] xy=\operatorname{diag}(x)y=(x_i y_i)_{i\in[n]} x y = diag ( x ) y = ( x i y i ) i ∈ [ n ] x ∘ y x \circ y x ∘ y x p x^p x p p ∈ Z p \in\mathbb{Z} p ∈ Z x \sqrt{x} x x / y x/y x / y x ∈ R n x \in\mathbb{R}^n x ∈ R n x ∈ R n x \in\mathbb{R}^n x ∈ R n p ∈ Z ≥ 1 ∪ { ∞ } p \in\mathbb{Z}_{\geq1}\cup\{\infty\} p ∈ Z ≥ 1 ∪ { ∞ } ∥ x ∥ p \lVert x\rVert_p ∥ x ∥ p ℓ p \ell_p ℓ p ℓ 2 \ell_2 ℓ 2 ∥ x ∥ \lVert x\rVert ∥ x ∥
Operator norms and vector norms. Any norm ∥ ⋅ ∥ \lVert\cdot\rVert ∥ ⋅ ∥ R n \mathbb{R}^n R n ∥ A ∥ = max x ≠ 0 ∥ A x ∥ ∥ x ∥ \lVert A\rVert=\max_{x\ne0}\frac{\lVert Ax\rVert}{\lVert x\rVert} ∥ A ∥ = max x = 0 ∥ x ∥ ∥ A x ∥ R n × n \mathbb{R}^{n\times n} R n × n ∥ A ∥ p → q = max x ≠ 0 ∥ A x ∥ q ∥ x ∥ p \lVert A\rVert_{p\to q}=\max_{x\ne0}\frac{\lVert Ax\rVert_q}{\lVert x\rVert_p} ∥ A ∥ p → q = max x = 0 ∥ x ∥ p ∥ A x ∥ q ℓ p → ℓ q \ell_p\to\ell_q ℓ p → ℓ q ∥ A ∥ max \lVert A\rVert_{\max} ∥ A ∥ m a x A A A
Feasibility and centrality. Let F ( M , q ) ⊆ R ≥ 0 2 n \mathcal{F}(M,q)\subseteq\mathbb{R}_{\geq0}^{2n} F ( M , q ) ⊆ R ≥ 0 2 n F > 0 ( M , q ) ⊆ R > 0 2 n \mathcal{F}_{>0}(M,q)\subseteq\mathbb{R}_{>0}^{2n} F > 0 ( M , q ) ⊆ R > 0 2 n
F ( M , q ) : = { ( x , s ) ∈ R ≥ 0 2 n : s = M x + q } , F > 0 ( M , q ) = F ( M , q ) ∩ R > 0 2 n . \mathcal{F}(M,q):=\{(x,s)\in\mathbb{R}_{\geq0}^{2n}:s=Mx+q\},\qquad\mathcal{F}_{>0}(M,q)=\mathcal{F}(M,q)\cap\mathbb{R}_{>0}^{2n}. F ( M , q ) := {( x , s ) ∈ R ≥ 0 2 n : s = M x + q } , F > 0 ( M , q ) = F ( M , q ) ∩ R > 0 2 n . Whenever clear from the context, we simply write F = F ( M , q ) \mathcal{F}=\mathcal{F}(M,q) F = F ( M , q ) F > 0 = F > 0 ( M , q ) \mathcal{F}_{>0}=\mathcal{F}_{>0}(M,q) F > 0 = F > 0 ( M , q ) ( x , s ) ∈ F ( M , q ) (x,s)\in\mathcal{F}(M,q) ( x , s ) ∈ F ( M , q ) feasible solution . A solution to LCP$(M,q)$ is called a complementary feasible solution or an LCP solution . Interestingly, for row sufficient matrices, LCP$(M,q)$ is solvable if and only if F ( M , q ) ≠ ∅ \mathcal{F}(M,q)\ne\varnothing F ( M , q ) = ∅ 3 [22 ] , see [32 ] , as well as [8 Section 3.5] .
Theorem 2.1. Let M ∈ R n × n M\in\mathbb{R}^{n\times n} M ∈ R n × n q ∈ R n q\in\mathbb{R}^n q ∈ R n F ( M , q ) ≠ ∅ \mathcal{F}(M,q)\ne\varnothing F ( M , q ) = ∅
The central path is defined as the set
C P : = { ( x , s ) ∈ F > 0 , x i s i = x ⊤ s n , ∀ i ∈ [ n ] } . \mathrm{CP}:=\left\{(x,s)\in\mathcal{F}_{>0},\ x_i s_i=\frac{x^\top s}{n},\ \forall i\in[n]\right\}. CP := { ( x , s ) ∈ F > 0 , x i s i = n x ⊤ s , ∀ i ∈ [ n ] } . For ( x , s ) ∈ F > 0 (x,s)\in\mathcal{F}_{>0} ( x , s ) ∈ F > 0 μ > 0 \mu>0 μ > 0 centrality error vector of ( x , s , μ ) (x,s,\mu) ( x , s , μ )
err ( x , s , μ ) : = 1 n − x s μ . (4) \operatorname{err}(x,s,\mu):=\mathbf{1}_n-\frac{xs}{\mu}. \tag*{(4)} err ( x , s , μ ) := 1 n − μ x s . (4) For β ≥ 0 \beta\geq0 β ≥ 0 ℓ 2 \ell_2 ℓ 2
N 2 ( β ) : = { ( x , s , μ ) : ( x , s ) ∈ F > 0 , μ ∈ R > 0 , ∥ err ( x , s , μ ) ∥ 2 ≤ β } . \mathcal{N}^2(\beta):=\{(x,s,\mu):(x,s)\in\mathcal{F}_{>0},\ \mu\in\mathbb{R}_{>0},\ \lVert\operatorname{err}(x,s,\mu)\rVert_2\leq\beta\}. N 2 ( β ) := {( x , s , μ ) : ( x , s ) ∈ F > 0 , μ ∈ R > 0 , ∥ err ( x , s , μ ) ∥ 2 ≤ β } . Background on P P P P ⋆ P_{\star} P ⋆ P 0 P_0 P 0
We recall that M ∈ P ⋆ M\in P_{\star} M ∈ P ⋆ M ∈ P ⋆ ( κ ) M\in P_{\star}(\kappa) M ∈ P ⋆ ( κ ) κ ≥ 0 \kappa\geq0 κ ≥ 0
( 1 + 4 κ ) ∑ i ∈ I M + ( x ) x i ( M x ) i + ∑ i ∈ I M − ( x ) x i ( M x ) i ≥ 0. (1+4\kappa)\sum_{i\in I_M^+(x)}x_i(Mx)_i+\sum_{i\in I_M^-(x)}x_i(Mx)_i\geq0. ( 1 + 4 κ ) i ∈ I M + ( x ) ∑ x i ( M x ) i + i ∈ I M − ( x ) ∑ x i ( M x ) i ≥ 0. The handicap number κ ^ ( M ) \widehat{\kappa}(M) κ ( M ) κ \kappa κ M ∈ P ⋆ M\in P_{\star} M ∈ P ⋆ ∞ \infty ∞ M ∉ P ⋆ M\notin P_{\star} M ∈ / P ⋆
Lemma 2.2. For any M ∈ P ⋆ ∩ R n × n M\in P_{\star}\cap\mathbb{R}^{n\times n} M ∈ P ⋆ ∩ R n × n D ∈ D > 0 n D\in D_{>0}^n D ∈ D > 0 n κ ^ ( D M D ) = κ ^ ( M ) \widehat{\kappa}(DMD)=\widehat{\kappa}(M) κ ( D M D ) = κ ( M )
One can also consider the handicap number of M M M x ∈ R n x\in\mathbb{R}^n x ∈ R n
κ ^ ( M , x ) : = { ∞ if x ⊤ M x < 0 and I M + ( x ) = ∅ , ∑ i ∈ I M − ( x ) x i ( M x ) i − 4 ∑ i ∈ I M + ( x ) x i ( M x ) i − 1 4 if x ⊤ M x < 0 and I M + ( x ) ≠ ∅ , 0 , if x ⊤ M x ≥ 0. (5) \widehat{\kappa}(M,x):=
\begin{cases}
\infty& \text{if }x^\top Mx<0\text{ and }I_M^+(x)=\varnothing,\\
\frac{\sum_{i\in I_M^-(x)}x_i(Mx)_i}{-4\sum_{i\in I_M^+(x)}x_i(Mx)_i}-\frac{1}{4} & \text{if }x^\top Mx<0\text{ and }I_M^+(x)\ne\varnothing,\\
0, & \text{if }x^\top Mx\geq0.
\end{cases} \tag*{(5)} κ ( M , x ) := ⎩ ⎨ ⎧ ∞ − 4 ∑ i ∈ I M + ( x ) x i ( M x ) i ∑ i ∈ I M − ( x ) x i ( M x ) i − 4 1 0 , if x ⊤ M x < 0 and I M + ( x ) = ∅ , if x ⊤ M x < 0 and I M + ( x ) = ∅ , if x ⊤ M x ≥ 0. (5) With this notation, we can equivalently write κ ^ ( M ) = sup x ∈ R n κ ^ ( M , x ) \hat{\kappa}(M)=\sup_{x\in\mathbb{R}^n}\hat{\kappa}(M,x) κ ^ ( M ) = sup x ∈ R n κ ^ ( M , x ) κ ^ ( M , x ) \hat{\kappa}(M,x) κ ^ ( M , x ) x x x P P P [82 ] .
Recall that M ∈ R n × n M\in\mathbb{R}^{n\times n} M ∈ R n × n x ∈ R n x\in\mathbb{R}^n x ∈ R n x ∘ ( M x ) ≤ 0 n x\circ(Mx)\leq\mathbf{0}_n x ∘ ( M x ) ≤ 0 n x ∘ ( M x ) = 0 n x\circ(Mx)=\mathbf{0}_n x ∘ ( M x ) = 0 n M T M^{\mathsf{T}} M T P ⋆ P_\star P ⋆
Theorem 2.3 (Väliaho [81 ] ). A matrix M ∈ R n × n M\in\mathbb{R}^{n\times n} M ∈ R n × n M ∈ P ⋆ M\in P_\star M ∈ P ⋆
The following lemma considers the relations and properties of P P P P ⋆ P_\star P ⋆ P 0 P_0 P 0 [8 Chapter 3] .
Lemma 2.4. We have P ⊂ P ⋆ ⊂ P 0 P\subset P_\star\subset P_0 P ⊂ P ⋆ ⊂ P 0 M ∈ P 0 M\in P_0 M ∈ P 0 M + ε I n ∈ P M+\varepsilon I_n\in P M + ε I n ∈ P ε > 0 \varepsilon>0 ε > 0 cl ( P ) = P 0 \operatorname{cl}(P)=P_0 cl ( P ) = P 0 M ∈ P M\in P M ∈ P M M M M − 1 ∈ P M^{-1}\in P M − 1 ∈ P
In particular, this justifies the invertibility in Theorem 1.1.
Corollary 2.5. For any M ∈ P 0 M\in P_0 M ∈ P 0 D ∈ D > 0 n D\in\mathcal{D}_{>0}^n D ∈ D > 0 n I n + D M D I_n+DMD I n + D M D
The following theorem by Väliaho which was originally established through more involved arguments in [82 ] , now follows as a corollary of Theorem 1.1.
Theorem 2.6 (Väliaho [82 ] ). For any M ∈ P ⋆ ∩ R n × n M\in P_\star\cap\mathbb{R}^{n\times n} M ∈ P ⋆ ∩ R n × n M T ∈ P ⋆ M^{\mathsf{T}}\in P_\star M T ∈ P ⋆ κ ^ ( M ) = κ ^ ( M T ) \hat{\kappa}(M)=\hat{\kappa}(M^{\mathsf{T}}) κ ^ ( M ) = κ ^ ( M T )
Proof. Immediate from Theorem 1.1, noting that sup D ∈ D > 0 n ∥ ( I n + D M D ) − 1 ∥ 2 = sup D ∈ D > 0 n ∥ ( I n + D M T D ) − 1 ∥ 2 \sup_{D\in\mathcal{D}_{>0}^n}\lVert(I_n+DMD)^{-1}\rVert_2=\sup_{D\in\mathcal{D}_{>0}^n}\lVert(I_n+DM^{\mathsf{T}}D)^{-1}\rVert_2 sup D ∈ D > 0 n ∥( I n + D M D ) − 1 ∥ 2 = sup D ∈ D > 0 n ∥( I n + D M T D ) − 1 ∥ 2
Rump’s characterizations of the sign-real spectral radius
Here we provide preliminaries necessary for understanding and operating with Rump’s characterization of the largest in magnitude real eigenvalue of a sufficient matrix. Let S n ⊆ R n × n \mathcal{S}^n\subseteq\mathbb{R}^{n\times n} S n ⊆ R n × n ± 1 \pm1 ± 1
Definition 2.7. For M ∈ R n × n M\in\mathbb{R}^{n\times n} M ∈ R n × n real spectral radius ρ ( M ) \rho(M) ρ ( M ) ρ ( M ) = 0 \rho(M)=0 ρ ( M ) = 0 sign-real spectral radius of M M M ρ 0 S ( M ) : = max S ∈ S n ρ ( S M ) \rho_0^{\mathcal{S}}(M):=\max_{S\in\mathcal{S}^n}\rho(SM) ρ 0 S ( M ) := max S ∈ S n ρ ( S M )
The following fundamental connection between sign-spectral radius and sufficient matrices is implied by Rump’s theorem [66 ] stated later as Theorem 7.4. We include the simple proof for completeness and for intuition.
Lemma 2.8. For A ∈ P 0 ∩ R n × n A\in P_0\cap\mathbb{R}^{n\times n} A ∈ P 0 ∩ R n × n ρ 0 S ( ( I n + A ) − 1 ) ≤ 1 \rho_0^{\mathcal{S}}((I_n+A)^{-1})\leq1 ρ 0 S (( I n + A ) − 1 ) ≤ 1
Proof. For a contradiction, assume ρ 0 S ( ( I n + A ) − 1 ) = λ > 1 \rho_0^{\mathcal{S}}((I_n+A)^{-1})=\lambda>1 ρ 0 S (( I n + A ) − 1 ) = λ > 1 S ∈ S n S\in\mathcal{S}^n S ∈ S n v ∈ R n ∖ { 0 } v\in\mathbb{R}^n\setminus\{0\} v ∈ R n ∖ { 0 } S ( I n + A ) − 1 v = λ v S(I_n+A)^{-1}v=\lambda v S ( I n + A ) − 1 v = λ v v ′ = S v v'=Sv v ′ = S v ( I n + A ) − 1 S v ′ = λ v ′ (I_n+A)^{-1}Sv'=\lambda v' ( I n + A ) − 1 S v ′ = λ v ′ ( I n + A ) v ′ = 1 λ S v ′ (I_n+A)v'=\frac{1}{\lambda}Sv' ( I n + A ) v ′ = λ 1 S v ′ ( D + A ) v ′ = 0 (D+A)v'=0 ( D + A ) v ′ = 0 D = I n − 1 λ S D=I_n-\frac{1}{\lambda}S D = I n − λ 1 S D D D λ \lambda λ D + A = D 1 / 2 ( I n + D − 1 / 2 A D − 1 / 2 ) D 1 / 2 D+A=D^{1/2}(I_n+D^{-1/2}AD^{-1/2})D^{1/2} D + A = D 1/2 ( I n + D − 1/2 A D − 1/2 ) D 1/2 Corollary 2.5 , a contradiction. □ \square □
A remarkable theorem by Rump relates the maximum absolute geometric cycle mean of a matrix to its sign-real spectral radius. In the context of matrices, a cycle takes the form C = ( i 1 , … , i ℓ , i ℓ + 1 = i 1 ) C=(i_1,\ldots,i_\ell,i_{\ell+1}=i_1) C = ( i 1 , … , i ℓ , i ℓ + 1 = i 1 ) i 1 , … , i ℓ i_1,\ldots,i_\ell i 1 , … , i ℓ [ n ] [n] [ n ] ∣ C ∣ = ℓ ≥ 1 |C|=\ell\geq1 ∣ C ∣ = ℓ ≥ 1 ℓ = 1 \ell=1 ℓ = 1 C = ( i , i ) C=(i,i) C = ( i , i ) i ∈ [ n ] i\in[n] i ∈ [ n ]
Definition 2.9. For a matrix M ∈ R n × n M\in\mathbb{R}^{n\times n} M ∈ R n × n M M M
ζ ( M ) : = max cycles C ( ∏ ( i , j ) ∈ C ∣ M i j ∣ ) 1 / ∣ C ∣ . \zeta(M):=\max_{\text{cycles }C}\left(\prod_{(i,j)\in C}|M_{ij}|\right)^{1/|C|}. ζ ( M ) := cycles C max ( i , j ) ∈ C ∏ ∣ M ij ∣ 1/∣ C ∣ . Rump [66 ] proved that for any M ∈ R n × n M\in\mathbb{R}^{n\times n} M ∈ R n × n ζ ( M ) \zeta(M) ζ ( M ) ρ 0 S ( M ) \rho_0^{\mathcal{S}}(M) ρ 0 S ( M ) 1 / n 1/n 1/ n ρ 0 S ( M ) \rho_0^{\mathcal{S}}(M) ρ 0 S ( M )
Theorem 2.10 ([66 Theorem 5.5] ). One has that
ζ ( M ) 3 + 2 2 ≤ ρ 0 S ( M ) ≤ n ⋅ ζ ( M ) . \frac{\zeta(M)}{3+2\sqrt{2}}\leq\rho_0^{\mathcal{S}}(M)\leq n\cdot\zeta(M). 3 + 2 2 ζ ( M ) ≤ ρ 0 S ( M ) ≤ n ⋅ ζ ( M ) . Maximum mean cycles
We use the following well-known result from combinatorial optimization. This is the multiplicative version of Gallai’s characterization of graphs with no negative weight cycles [23 ] , see e.g. [1 Chapter 5] . Such a multiplicative version is commonly used in the context of generalized flows, see e.g. [76 ] for an early example, and also [1 Chapter 15] .
Theorem 2.11. Let G = ( V , E , ρ ) G=(V,E,\rho) G = ( V , E , ρ ) R ≥ 1 R\ge1 R ≥ 1 ρ ( i , j ) ∈ [ 0 , R ] \rho(i,j)\in[0,R] ρ ( i , j ) ∈ [ 0 , R ] i ≠ j i\ne j i = j ∏ e ∈ E ( C ) ρ ( e ) ≤ 1 \prod_{e\in E(C)}\rho(e)\le1 ∏ e ∈ E ( C ) ρ ( e ) ≤ 1 C C C
π j : = max { ∏ e ∈ E ( P ) ρ ( e ) : P is a directed walk ending in j } , ∀ j ∈ V , (6) \pi_j:=\max\left\{\prod_{e\in E(P)}\rho(e):P\text{ is a directed walk ending in }j\right\},\quad\forall j\in V, \tag*{(6)} π j := max ⎩ ⎨ ⎧ e ∈ E ( P ) ∏ ρ ( e ) : P is a directed walk ending in j ⎭ ⎬ ⎫ , ∀ j ∈ V , (6) with the convention that the product is 1 1 1 π j \pi_j π j π j ∈ [ 1 , R ∣ V ∣ ] \pi_j\in[1,R^{|V|}] π j ∈ [ 1 , R ∣ V ∣ ]
π i ρ ( i , j ) ≤ π j ∀ ( i , j ) ∈ E . \pi_i\rho(i,j)\le\pi_j\quad\forall(i,j)\in E. π i ρ ( i , j ) ≤ π j ∀ ( i , j ) ∈ E . Moreover, π \pi π O ( ∣ V ∣ 3 ) O(|V|^3) O ( ∣ V ∣ 3 )
In the context of Theorem 2.10 , we get the following. We note that such rescalings have been independently introduced in the linear algebra context by Fiedler and Pták [21 ] , see also [30 ] .
Corollary 2.12. Let H ∈ R n × n H\in\mathbb{R}^{n\times n} H ∈ R n × n ζ ( H ) ≤ τ \zeta(H)\le\tau ζ ( H ) ≤ τ ∣ H i i ∣ ≤ τ |H_{ii}|\le\tau ∣ H ii ∣ ≤ τ i ∈ [ n ] i\in[n] i ∈ [ n ] τ > 0 \tau>0 τ > 0 q H q_H q H q H ( i ) : = π i q_H(i):=\pi_i q H ( i ) := π i (6) for the weights ρ ( i , j ) : = ∣ H i j ∣ / τ \rho(i,j):=|H_{ij}|/\tau ρ ( i , j ) := ∣ H ij ∣/ τ Q H = diag ( q H ) Q_H=\operatorname{diag}(q_H) Q H = diag ( q H ) ∥ Q H H Q H − 1 ∥ max ≤ τ \lVert Q_HHQ_H^{-1}\rVert_{\max}\le\tau ∥ Q H H Q H − 1 ∥ m a x ≤ τ q H ( i ) q_H(i) q H ( i ) (6) is taken along a simple path.
Overview of approach
Here we provide an overview of our overall approach. We split our approach for solving P ⋆ \mathcal{P}_{\star} P ⋆ § 3.1 we describe a basic IPM which converges in time proportional to the handicap number κ ^ ( M ) \hat{\kappa}(M) κ ^ ( M ) § 3.2 we show how Rump’s theorem (Theorem 2.10) asserts the existence of a positive diagonal similarity rescaling Q Q Q M M M I n + Q M Q − 1 I_n+QMQ^{-1} I n + QM Q − 1 § 3.3 in which we outline our strategy to prove the stability of Q Q Q
We also highlight the key steps for the two hardness results. In § 3.4 we describe how Rump’s characterization of P \mathcal{P} P M ~ \widetilde{M} M M M M M ∈ P M\in\mathcal{P} M ∈ P M ~ ∈ P ⋆ \widetilde{M}\in\mathcal{P}_{\star} M ∈ P ⋆
Basic IPM overview
Throughout, let us assume M ∈ P κ ^ M\in\mathcal{P}_{\hat{\kappa}} M ∈ P κ ^ x x x s = M x + q s=Mx+q s = M x + q
x ( M x + q ) = μ 1 n x(Mx+q)=\mu\mathbf{1}_n x ( M x + q ) = μ 1 n with x , s > 0 x,s>0 x , s > 0 μ \mu μ x x x μ \mu μ x x x x ( M x + q ) = w x(Mx+q)=w x ( M x + q ) = w w ∈ R > 0 n w\in\mathbb{R}^{n}_{>0} w ∈ R > 0 n x → x + Δ x x\to x+\Delta x x → x + Δ x
( x + Δ x ) ( M ( x + Δ x ) + q ) ≈ μ 1 n . (7) (x+\Delta x)(M(x+\Delta x)+q)\approx\mu\mathbf{1}_n. \tag*{(7)} ( x + Δ x ) ( M ( x + Δ x ) + q ) ≈ μ 1 n . (7) To find a good update direction, one may simply linearize the left-hand side of (7) in Δ x \Delta x Δ x diag ( w ) \operatorname{diag}(w) diag ( w ) μ I n \mu I_n μ I n
( μ I n + X M X ) Δ x x = μ 1 n − w . (\mu I_n+XMX)\frac{\Delta x}{x}=\mu\mathbf{1}_n-w. ( μ I n + X M X ) x Δ x = μ 1 n − w . To see how this makes progress, we can explicitly calculate the new centrality error by directly expanding
( x + Δ x ) ( M ( x + Δ x ) + q ) − μ 1 n = ( 1 n + Δ x x ) ( w + X M Δ x ) − μ 1 n = ( 1 n + Δ x x ) μ 1 n ( 1 n − Δ x x ) − μ 1 n = − μ 1 n ( Δ x x ) 2 . (x+\Delta x)(M(x+\Delta x)+q)-\mu1_n=\left(1_n+\frac{\Delta x}{x}\right)(w+XM\Delta x)-\mu1_n
=\left(1_n+\frac{\Delta x}{x}\right)\mu1_n\left(1_n-\frac{\Delta x}{x}\right)-\mu1_n
=-\mu1_n\left(\frac{\Delta x}{x}\right)^2. ( x + Δ x ) ( M ( x + Δ x ) + q ) − μ 1 n = ( 1 n + x Δ x ) ( w + X M Δ x ) − μ 1 n = ( 1 n + x Δ x ) μ 1 n ( 1 n − x Δ x ) − μ 1 n = − μ 1 n ( x Δ x ) 2 . Thus, recalling (4) , the centrality error vector changes from err : = 1 n − w μ \operatorname{err}:=1_n-\frac{w}{\mu} err := 1 n − μ w
err ′ = ( Δ x x ) 2 = ( ( I n + 1 μ X M X ) − 1 err ) 2 . (8) \operatorname{err}'=\left(\frac{\Delta x}{x}\right)^2=\left(\left(I_n+\frac{1}{\mu}XMX\right)^{-1}\operatorname{err}\right)^2. \tag*{(8)} err ′ = ( x Δ x ) 2 = ( ( I n + μ 1 X M X ) − 1 err ) 2 . (8) The same expansion shows that the update preserves ( x + Δ x , M ( x + Δ x ) + q ) ∈ F > 0 (x+\Delta x,M(x+\Delta x)+q)\in\mathcal{F}_{>0} ( x + Δ x , M ( x + Δ x ) + q ) ∈ F > 0 ∥ Δ x / x ∥ ∞ < 1 \|\Delta x/x\|_{\infty}<1 ∥Δ x / x ∥ ∞ < 1 (8) captures the quadratic convergence of Newton’s method, by showing that error contracts very quickly, provided it was small enough to begin with. For example, if M M M ∥ err ′ ∥ 2 ≤ ∥ err ∥ 2 2 \|\operatorname{err}'\|_2\leq\|\operatorname{err}\|_2^2 ∥ err ′ ∥ 2 ≤ ∥ err ∥ 2 2 ∥ err ∥ 2 ≤ 1 / 2 \|\operatorname{err}\|_2\leq1/2 ∥ err ∥ 2 ≤ 1/2
While this inequality need not hold for a general P ⋆ P_{\star} P ⋆ (8) by O ( 1 + κ ) O(\sqrt{1+\kappa}) O ( 1 + κ )
∥ err ′ ∥ 2 2 = ∥ ( I n + 1 μ X M X ) − 1 err ∥ 4 4 ≤ ∥ ( I n + 1 μ X M X ) − 1 err ∥ 2 2 ≤ ∥ ( I n + 1 μ X M X ) − 1 ∥ 2 2 ∥ err ∥ 2 2 ≤ O ( 1 + κ ) ∥ err ∥ 2 2 . \begin{aligned}
\|\operatorname{err}'\|_2^2=\left\|\left(I_n+\frac{1}{\mu}XMX\right)^{-1}\operatorname{err}\right\|_4^4\leq\left\|\left(I_n+\frac{1}{\mu}XMX\right)^{-1}\operatorname{err}\right\|_2^2 \\
&\leq\left\|\left(I_n+\frac{1}{\mu}XMX\right)^{-1}\right\|_2^2\|\operatorname{err}\|_2^2\leq O(1+\kappa)\|\operatorname{err}\|_2^2.
\end{aligned} ∥ err ′ ∥ 2 2 = ( I n + μ 1 X M X ) − 1 err 4 4 ≤ ( I n + μ 1 X M X ) − 1 err 2 2 ≤ ( I n + μ 1 X M X ) − 1 2 2 ∥ err ∥ 2 2 ≤ O ( 1 + κ ) ∥ err ∥ 2 2 . Thus, if initially ∥ err ∥ 2 ≤ c / ( 1 + κ ) \|\operatorname{err}\|_2\leq c/(1+\kappa) ∥ err ∥ 2 ≤ c / ( 1 + κ ) c > 0 c>0 c > 0
With this in mind, we can see how the handicap number affects the convergence. Imagine that x x x μ 0 \mu_0 μ 0 x x x μ = μ 0 / ( 1 + δ ) \mu=\mu_0/(1+\delta) μ = μ 0 / ( 1 + δ ) δ > 0 \delta>0 δ > 0 μ \mu μ err = 1 n − μ 0 μ 1 n = − δ 1 n \operatorname{err}=1_n-\frac{\mu_0}{\mu}1_n=-\delta1_n err = 1 n − μ μ 0 1 n = − δ 1 n ∥ err ∥ 2 = δ n 1 / 2 ≤ c / ( 1 + κ ) \|\operatorname{err}\|_2=\delta n^{1/2}\leq c/(1+\kappa) ∥ err ∥ 2 = δ n 1/2 ≤ c / ( 1 + κ ) δ = c / ( n ( 1 + κ ) ) \delta=c/(\sqrt{n}(1+\kappa)) δ = c / ( n ( 1 + κ )) O ~ ( n ( 1 + κ ) ) \widetilde{O}(\sqrt{n}(1+\kappa)) O ( n ( 1 + κ ))
Improving the convergence analysis via Rump weights
The preceding analysis appears to make it difficult to improve the dependence on κ \kappa κ ∥ ( I n + 1 μ X M X ) − 1 ∥ \|(I_n+\frac{1}{\mu}XMX)^{-1}\| ∥ ( I n + μ 1 X M X ) − 1 ∥ worst-case scaling X X X M M M x x x M M M
While this does not give a priori reason to yield any faster convergence, we notice that a remarkable theorem of Rump (Theorem 2.10, c.f. [66 ] ) asserts the existence of a similarity rescaling of H = ( I n + 1 μ X M X ) − 1 H=(I_n+\frac{1}{\mu}XMX)^{-1} H = ( I n + μ 1 X M X ) − 1 Q H = diag ( q H ) Q_H=\operatorname{diag}(q_H) Q H = diag ( q H )
This suggests changing the metric used to measure centrality error so that the contraction depends on ∥ Q H H Q H − 1 ∥ 2 \|Q_HHQ_H^{-1}\|_2 ∥ Q H H Q H − 1 ∥ 2 ∥ H ∥ 2 \|H\|_2 ∥ H ∥ 2 ∥ Q H x ∥ 2 \|Q_Hx\|_2 ∥ Q H x ∥ 2 q H q_H q H Rump weights , and the resulting norm the Rump norm .
Following the same idea as before, let us see how the error changes after a Newton step when measured in Rump norm. Assuming q H ≥ 1 n q_H \ge1_n q H ≥ 1 n
∥ Q H e r r ′ ∥ 2 2 = ∥ Q H ( H e r r ) 2 ∥ 2 ≤ n 1 / 2 ∥ Q H H e r r ∥ ∞ 2 = n 1 / 2 ∥ Q H H Q H − 1 ⋅ Q H e r r ∥ ∞ 2 ≤ n 1 / 2 ∥ Q H H Q H − 1 ∥ 2 → ∞ 2 ∥ Q H e r r ∥ 2 2 ≤ O ( n 3 / 2 ) ⋅ ∥ Q H e r r ∥ 2 2 . (9) \begin{aligned}
\lVert Q_H\mathrm{err}'\rVert_2^2 &= \lVert Q_H(H\mathrm{err})^2\rVert_2 \\
&\le n^{1/2}\lVert Q_HH\mathrm{err}\rVert_\infty^2 \\
&= n^{1/2}\lVert Q_HHQ_H^{-1}\cdot Q_H\mathrm{err}\rVert_\infty^2 \\
&\le n^{1/2}\lVert Q_HHQ_H^{-1}\rVert_{2\to\infty}^2\lVert Q_H\mathrm{err}\rVert_2^2 \\
&\le O(n^{3/2})\cdot\lVert Q_H\mathrm{err}\rVert_2^2. \tag*{(9)}
\end{aligned} ∥ Q H err ′ ∥ 2 2 = ∥ Q H ( H err ) 2 ∥ 2 ≤ n 1/2 ∥ Q H H err ∥ ∞ 2 = n 1/2 ∥ Q H H Q H − 1 ⋅ Q H err ∥ ∞ 2 ≤ n 1/2 ∥ Q H H Q H − 1 ∥ 2 → ∞ 2 ∥ Q H err ∥ 2 2 ≤ O ( n 3/2 ) ⋅ ∥ Q H err ∥ 2 2 . (9) where the last inequality follows from ∥ Q H H Q H − 1 ∥ 2 → ∞ = O ( n 1 / 2 ) \lVert Q_HHQ_H^{-1}\rVert_{2\to\infty}=O(n^{1/2}) ∥ Q H H Q H − 1 ∥ 2 → ∞ = O ( n 1/2 ) Q H H Q H − 1 Q_HHQ_H^{-1} Q H H Q H − 1 ∥ Q H e r r ∥ 2 = O ( 1 / n 3 / 2 ) \lVert Q_H\mathrm{err}\rVert_2=O(1/n^{3/2}) ∥ Q H err ∥ 2 = O ( 1/ n 3/2 )
There are two remaining challenges.
First, the Rump weights depend on the matrix H H H x x x μ \mu μ x x x x ′ x' x ′ H ′ H' H ′ q H ′ q_{H'} q H ′ (9) . Indeed, in Lemma 5.7 we show that there exist Rump weights which are sufficiently stable under Newton steps in the sense that q H ′ ≤ 2 q H q_{H'}\le2q_H q H ′ ≤ 2 q H (9) while changing from Q H Q_H Q H Q H ′ Q_{H'} Q H ′ ∥ Q H ′ e r r ′ ∥ 2 ≤ O ( n 3 / 2 ) ∥ Q H e r r ∥ 2 2 \lVert Q_{H'}\mathrm{err}'\rVert_2\le O(n^{3/2})\lVert Q_H\mathrm{err}\rVert_2^2 ∥ Q H ′ err ′ ∥ 2 ≤ O ( n 3/2 ) ∥ Q H err ∥ 2 2
Second, as a result of (9) , Newton’s method is guaranteed to converge provided that the Rump norm of the error satisfies ∥ Q H e r r ∥ 2 ≤ c / n 3 / 2 \lVert Q_H\mathrm{err}\rVert_2\le c/n^{3/2} ∥ Q H err ∥ 2 ≤ c / n 3/2 c > 0 c>0 c > 0
Formally, assuming that we start from exact centrality, after we dial down the target centrality parameter by a factor of 1 + δ 1+\delta 1 + δ ∥ Q H e r r ∥ = ∥ Q H ⋅ δ 1 n ∥ 2 = δ ∥ q H ∥ 2 \lVert Q_H\mathrm{err}\rVert=\lVert Q_H\cdot\delta1_n\rVert_2=\delta\lVert q_H\rVert_2 ∥ Q H err ∥ = ∥ Q H ⋅ δ 1 n ∥ 2 = δ ∥ q H ∥ 2 H H H Lemma 5.6 bounds Rump weights by
1 n ≤ q H ≤ O ( 1 + κ ) 1 n . 1_n\le q_H\le O(\sqrt{1+\kappa})1_n. 1 n ≤ q H ≤ O ( 1 + κ ) 1 n . Thus ∥ q H ∥ 2 ≤ O ( n 1 / 2 1 + κ ) \lVert q_H\rVert_2\le O(n^{1/2}\sqrt{1+\kappa}) ∥ q H ∥ 2 ≤ O ( n 1/2 1 + κ ) δ ≈ 1 n 2 1 + κ \delta\approx\frac{1}{n^2\sqrt{1+\kappa}} δ ≈ n 2 1 + κ 1 O ( n 2 1 + κ ) O(n^2\sqrt{1+\kappa}) O ( n 2 1 + κ ) μ \mu μ
For the first one, instead of setting δ \delta δ δ ≈ 1 n 5 / 4 1 + κ \delta\approx\frac{1}{n^{5/4}\sqrt{1+\kappa}} δ ≈ n 5/4 1 + κ 1 O ( n − 3 / 2 ) O(n^{-3/2}) O ( n − 3/2 ) δ ≈ 1 n 9 / 8 1 + κ \delta\approx\frac{1}{n^{9/8}\sqrt{1+\kappa}} δ ≈ n 9/8 1 + κ 1 O ( n − 3 / 2 ) O(n^{-3/2}) O ( n − 3/2 )
The complete analysis that achieves O ~ ( n 5 / 4 1 + κ ) \widetilde{O}(n^{5/4}\sqrt{1+\kappa}) O ( n 5/4 1 + κ ) § 5 . The further improvement to O ~ ( n 9 / 8 1 + κ ) \widetilde{O}(n^{9/8}\sqrt{1+\kappa}) O ( n 9/8 1 + κ )
Bounding Rump weights and proving stability
As we saw, the analysis hinges on proving the existence of Rump weights that are bounded and do not change too aggressively when we update x x x sign-real spectral radius , denoted by ρ 0 S \rho_0^S ρ 0 S Theorem 2.10 enables us to connect matrix sufficiency to combinatorial properties of H H H ∣ H ∣ |H| ∣ H ∣
First, Lemma 2.8 shows that for any sufficient matrix A A A H = ( I n + A ) − 1 H=(I_n+A)^{-1} H = ( I n + A ) − 1 1 1 1 ∣ H i j ∣ / K |H_{ij}|/K ∣ H ij ∣/ K K = 8 K=8 K = 8 3 + 2 2 K < 3 4 \frac{3+2\sqrt{2}}{K}<\frac{3}{4} K 3 + 2 2 < 4 3 Q H Q_H Q H ∣ Q H H Q H − 1 ∣ |Q_HHQ_H^{-1}| ∣ Q H H Q H − 1 ∣ K K K Q H Q_H Q H Rump weights and can be explicitly computed by letting q H ( j ) q_H(j) q H ( j ) j j j Definition 5.2 ).
Next, we need to understand how these path products change when we update x x x x x x Δ x ˉ = Δ x / x \Delta\bar{x} = \Delta x/x Δ x ˉ = Δ x / x U = diag ( 1 + Δ x ˉ ) U = \operatorname{diag}(1 + \Delta\bar{x}) U = diag ( 1 + Δ x ˉ ) A = μ − 1 X M X A = \mu^{-1}XMX A = μ − 1 X M X H ′ = ( I + U A U ) − 1 H' = (I + UAU)^{-1} H ′ = ( I + U A U ) − 1
∣ ( Q H H ′ Q H − 1 ) u v ∣ ≤ K exp ( O ( ∥ Δ x ˉ ∥ 1 ) ) . \left|(Q_H H' Q_H^{-1})_{uv}\right| \le K \exp\left(O(\lVert\Delta\bar{x}\rVert_1)\right). ( Q H H ′ Q H − 1 ) uv ≤ K exp ( O (∥ Δ x ˉ ∥ 1 ) ) . Now look at a simple path P P P i i i j j j q H ′ ( j ) q_{H'}(j) q H ′ ( j )
q H ′ ( j ) = ∏ ( u , v ) ∈ P ∣ H u v ′ ∣ K ≤ q H ( j ) q H ( i ) exp ( O ( ∣ P ∣ ∥ Δ x ˉ ∥ 1 ) ) ≤ q H ( j ) exp ( O ( n ∥ Δ x ˉ ∥ 1 ) ) . q_{H'}(j) = \prod_{(u,v)\in P} \frac{|H'_{uv}|}{K} \le\frac{q_H(j)}{q_H(i)}\exp\left(O(|P|\lVert\Delta\bar{x}\rVert_1)\right) \le q_H(j)\exp\left(O(n\lVert\Delta\bar{x}\rVert_1)\right). q H ′ ( j ) = ( u , v ) ∈ P ∏ K ∣ H uv ′ ∣ ≤ q H ( i ) q H ( j ) exp ( O ( ∣ P ∣ ∥ Δ x ˉ ∥ 1 ) ) ≤ q H ( j ) exp ( O ( n ∥ Δ x ˉ ∥ 1 ) ) . since q H ( i ) ≥ 1 q_H(i) \ge1 q H ( i ) ≥ 1 n n n Lemma 5.7 removes the factor of n n n
∥ log q H ′ − log q H ∥ ∞ ≤ 16 K ∥ log ( 1 + Δ x ˉ ) ∥ 1 = O ( ∥ Δ x ˉ ∥ 1 ) \lVert\log q_{H'}-\log q_H\rVert_\infty\le16K\lVert\log(1+\Delta\bar{x})\rVert_1 = O(\lVert\Delta\bar{x}\rVert_1) ∥ log q H ′ − log q H ∥ ∞ ≤ 16 K ∥ log ( 1 + Δ x ˉ ) ∥ 1 = O (∥ Δ x ˉ ∥ 1 ) when ∥ Δ x ˉ ∥ ∞ ≤ 1 / 2 \lVert\Delta\bar{x}\rVert_\infty\le1/2 ∥ Δ x ˉ ∥ ∞ ≤ 1/2
This suffices to show stability under corrector steps. Using Δ x ˉ = H e r r \Delta\bar{x}=H\mathit{err} Δ x ˉ = H err q H ≥ 1 n q_H\ge\mathbf{1}_n q H ≥ 1 n ∥ Q H H Q H − 1 ∥ 2 → ∞ = O ( n 1 / 2 ) \lVert Q_HHQ_H^{-1}\rVert_{2\to\infty}=O(n^{1/2}) ∥ Q H H Q H − 1 ∥ 2 → ∞ = O ( n 1/2 )
∥ Δ x ˉ ∥ 1 ≤ ∥ Q H H e r r ∥ 1 ≤ O ( n 3 / 2 ) ∥ Q H e r r ∥ 2 . \lVert\Delta\bar{x}\rVert_1 \le\lVert Q_HH\mathit{err}\rVert_1 \le O(n^{3/2})\lVert Q_H\mathit{err}\rVert_2. ∥ Δ x ˉ ∥ 1 ≤ ∥ Q H H err ∥ 1 ≤ O ( n 3/2 ) ∥ Q H err ∥ 2 . Thus, if ∥ Q H e r r ∥ ≤ c / n 3 / 2 \lVert Q_H\mathit{err}\rVert\le c/n^{3/2} ∥ Q H err ∥ ≤ c / n 3/2 c c c q H ′ ≤ 2 q H q_{H'}\le2q_H q H ′ ≤ 2 q H
Finally, to bound the weights, consider a simple path P : j → ⋯ → i P:j\to\cdots\to i P : j → ⋯ → i q H ( i ) > 1 q_H(i)>1 q H ( i ) > 1 j j j H H H z z z j j j i → j i\to j i → j C C C K K K 1 1 1
q H ( i ) ∣ z i ∣ K ≤ 1. q_H(i)\frac{|z_i|}{K}\le1. q H ( i ) K ∣ z i ∣ ≤ 1. Thus, we need to understand how large a vector z z z τ = 1 + κ \tau=\sqrt{1+\kappa} τ = 1 + κ v v v Lemma 5.12 allows the choice z = 2 H v K τ z=\frac{2Hv}{K\tau} z = K τ 2 H v ρ 0 S ( C ) ≤ 4 / 3 \rho_0^S(C)\le4/3 ρ 0 S ( C ) ≤ 4/3 ( 4 / 3 ) ( 3 + 2 2 ) < K (4/3)(3+2\sqrt{2})<K ( 4/3 ) ( 3 + 2 2 ) < K
q H ( i ) ∣ ( H v ) i ∣ ≤ K 2 2 τ . q_H(i)|(Hv)_i|\le\frac{K^2}{2}\tau. q H ( i ) ∣ ( H v ) i ∣ ≤ 2 K 2 τ . When q H ( i ) = 1 q_H(i)=1 q H ( i ) = 1 ∥ H ∥ 2 ≤ τ \lVert H\rVert_2\le\tau ∥ H ∥ 2 ≤ τ k → i k\to i k → i K q H ( i ) = q H ( k ) ∣ H k i ∣ Kq_H(i)=q_H(k)|H_{ki}| K q H ( i ) = q H ( k ) ∣ H k i ∣ k k k v = e i v=e_i v = e i
1 n ≤ q H ≤ K 2 1 + κ ⋅ 1 n , \mathbf{1}_n\le q_H\le\frac{K}{2}\sqrt{1+\kappa}\cdot\mathbf{1}_n, 1 n ≤ q H ≤ 2 K 1 + κ ⋅ 1 n , as claimed in Lemma 5.6 . Throughout the analysis, we use Lemma 2.2, which ensures that the matrix A = 1 μ X M X A=\frac{1}{\mu}XMX A = μ 1 X M X M M M § 5.4 and § 5.5 .
Hardness results
We recall that the sign-real spectral radius ρ 0 S ( M ) \rho_0^S(M) ρ 0 S ( M ) M ∈ R n × n M\in\mathbb{R}^{n\times n} M ∈ R n × n S M SM S M S S S ± 1 \pm1 ± 1 [66 ] ), P P P M M M P P P ρ 0 S ( M ) < 1 \rho_0^S(M)<1 ρ 0 S ( M ) < 1 C ( M ) : = ( I n − M ) − 1 ( I n + M ) C(M):=(I_n-M)^{-1}(I_n+M) C ( M ) := ( I n − M ) − 1 ( I n + M ) I n − M I_n-M I n − M § 7 .
Rump’s theorem can be extended in two ways. For a rational matrix M M M ρ 0 S ( M ) \rho_0^S(M) ρ 0 S ( M ) 1 1 1 M M M M ∈ Q n × n M\in\mathbb{Q}^{n\times n} M ∈ Q n × n M M M P P P ρ 0 S ( M ) ≤ 1 − ϵ \rho_0^S(M)\le1-\epsilon ρ 0 S ( M ) ≤ 1 − ϵ ϵ ∈ Q > 0 \epsilon\in\mathbb{Q}_{>0} ϵ ∈ Q > 0 length ( M ) \operatorname{length}(M) length ( M ) Lemma 2.4 , any P 0 P_0 P 0 P P P P 0 P_0 P 0 P P P P 0 P_0 P 0 M M M P 0 P_0 P 0 ρ 0 S ( M ) ≤ 1 \rho_0^S(M)\le1 ρ 0 S ( M ) ≤ 1 ρ 0 S ( ⋅ ) \rho_0^S(\cdot) ρ 0 S ( ⋅ ) ρ 0 S ( M t ) < 1 \rho_0^S(M_t)<1 ρ 0 S ( M t ) < 1
Overview of the proof of Theorem 1.3. We will exploit the aforementioned guarantee of the rational margin between the sign-real spectral radius and 1. For any given rational matrix M M M N N N M M M Proposition 7.3 for its definition and existence. We know that M ∈ P M \in\mathcal{P} M ∈ P ρ 0 S ( N ) < 1 \rho_0^S(N) < 1 ρ 0 S ( N ) < 1 α ∈ Q \alpha\in\mathbb{Q} α ∈ Q ρ 0 S ( N ) \rho_0^S(N) ρ 0 S ( N ) M ∈ P M \in\mathcal{P} M ∈ P M ∈ P M \in\mathcal{P} M ∈ P ρ 0 S ( α N ) = α ρ 0 S ( N ) < 1 \rho_0^S(\alpha N) = \alpha\rho_0^S(N) < 1 ρ 0 S ( α N ) = α ρ 0 S ( N ) < 1 M ∉ P M \notin\mathcal{P} M ∈ / P ρ 0 S ( α N ) = α ρ 0 S ( N ) ≥ α > 1 \rho_0^S(\alpha N) = \alpha\rho_0^S(N) \ge\alpha> 1 ρ 0 S ( α N ) = α ρ 0 S ( N ) ≥ α > 1 Claim 7.8 for the precise statement and its proof.
Now, setting M ~ \widetilde{M} M α N \alpha N α N ρ 0 S ( α N ) < 1 \rho_0^S(\alpha N) < 1 ρ 0 S ( α N ) < 1 M ~ ∈ P \widetilde{M} \in\mathcal{P} M ∈ P ρ 0 S ( α N ) > 1 \rho_0^S(\alpha N) > 1 ρ 0 S ( α N ) > 1 M ~ ∉ P 0 \widetilde{M} \notin\mathcal{P}_0 M ∈ / P 0 P 0 \mathcal{P}_0 P 0 P ~ \widetilde{\mathcal{P}} P P \mathcal{P} P P 0 \mathcal{P}_0 P 0 P ⊆ P ~ ⊆ P 0 \mathcal{P} \subseteq\widetilde{\mathcal{P}} \subseteq\mathcal{P}_0 P ⊆ P ⊆ P 0 M ~ ∈ P ~ \widetilde{M} \in\widetilde{\mathcal{P}} M ∈ P ρ 0 S ( α N ) < 1 \rho_0^S(\alpha N) < 1 ρ 0 S ( α N ) < 1 M ~ ∉ P ~ \widetilde{M} \notin\widetilde{\mathcal{P}} M ∈ / P ρ 0 S ( α N ) > 1 \rho_0^S(\alpha N) > 1 ρ 0 S ( α N ) > 1 ρ 0 S ( α N ) < 1 \rho_0^S(\alpha N) < 1 ρ 0 S ( α N ) < 1 M ∈ P M \in\mathcal{P} M ∈ P ρ 0 S ( α N ) > 1 \rho_0^S(\alpha N) > 1 ρ 0 S ( α N ) > 1 M ∉ P M \notin\mathcal{P} M ∈ / P M ~ ∈ P ~ \widetilde{M} \in\widetilde{\mathcal{P}} M ∈ P M ∈ P M \in\mathcal{P} M ∈ P Theorem 1.3 follows from the co-NP completeness of testing whether a rational matrix is in P \mathcal{P} P M ~ \widetilde{M} M
Overview of the proof of Theorem 1.4. The proof of Theorem 1.3 also indicates that there should be a connection between the sign-real spectral radius and handicap number. Rump proved that max s 1 , s 2 ∈ { − 1 , 1 } n s 1 T M s 2 \max_{s_1,s_2\in\{-1,1\}^n}s_1^{\mathsf T}Ms_2 max s 1 , s 2 ∈ { − 1 , 1 } n s 1 T M s 2 M M M [66 Theorem 2.8] , and the quantity max s 1 , s 2 ∈ { − 1 , 1 } n s 1 T M s 2 \max_{s_1,s_2\in\{-1,1\}^n}s_1^{\mathsf T}Ms_2 max s 1 , s 2 ∈ { − 1 , 1 } n s 1 T M s 2 [59 ] . Here, we reduce Max-Cut to handicap number computation, by constructing a P \mathcal{P} P
In Lemma 7.13 for any connected graph G = ( V , E ) G=(V,E) G = ( V , E ) ∣ V ∣ = n ≥ 2 \lvert V\rvert=n\ge2 ∣ V ∣ = n ≥ 2 ∣ E ∣ = m \lvert E\rvert=m ∣ E ∣ = m M G M_G M G A G ∈ { 0 , ± 1 } n × m A_G\in\{0,\pm1\}^{n\times m} A G ∈ { 0 , ± 1 } n × m G G G κ ^ ( M G ) = maxcut ( G ) − 1 4 \hat{\kappa}(M_G)=\operatorname{maxcut}(G)-\frac{1}{4} κ ^ ( M G ) = maxcut ( G ) − 4 1 A G A G T ∈ R n × n A_GA_G^{\mathsf T}\in\mathbb{R}^{n\times n} A G A G T ∈ R n × n maxcut ( G ) = 1 4 max v ∈ { − 1 , 1 } n ∥ A G T v ∥ 2 2 \operatorname{maxcut}(G)=\frac{1}{4}\max_{v\in\{-1,1\}^n}\lVert A_G^{\mathsf T}v\rVert_2^2 maxcut ( G ) = 4 1 max v ∈ { − 1 , 1 } n ∥ A G T v ∥ 2 2
M G : = ( I m 2 A G T 0 0 I n 21 n 0 0 1 ) ∈ Z ( m + n + 1 ) × ( m + n + 1 ) . M_G:=\begin{pmatrix}
I_m & 2A_G^{\mathsf T} & 0\\
0 & I_n & 2\mathbf{1}_n\\
0 & 0 & 1
\end{pmatrix}\in\mathbb{Z}^{(m+n+1)\times(m+n+1)}. M G := I m 0 0 2 A G T I n 0 0 2 1 n 1 ∈ Z ( m + n + 1 ) × ( m + n + 1 ) . We use the variable transformation y = ( x + M G x ) / 2 y=(x+M_Gx)/2 y = ( x + M G x ) /2 N = ( M G − I m + n + 1 ) / 2 N=(M_G-I_{m+n+1})/2 N = ( M G − I m + n + 1 ) /2 A G T 1 n = 0 A_G^{\mathsf T}\mathbf{1}_n=0 A G T 1 n = 0 N N N
x T M G x = ∥ y ∥ 2 2 − ∥ N ( I m + n + 1 + N ) − 1 y ∥ 2 2 = ∥ y 1 ∥ 2 2 + ∥ y 2 ∥ 2 2 − ∥ A G T y 2 ∥ 2 2 − ( n − 1 ) y 3 2 . (10) x^{\mathsf T}M_Gx=\lVert y\rVert_2^2-\lVert N(I_{m+n+1}+N)^{-1}y\rVert_2^2=\lVert y_1\rVert_2^2+\lVert y_2\rVert_2^2-\lVert A_G^{\mathsf T}y_2\rVert_2^2-(n-1)y_3^2. \tag*{(10)} x T M G x = ∥ y ∥ 2 2 − ∥ N ( I m + n + 1 + N ) − 1 y ∥ 2 2 = ∥ y 1 ∥ 2 2 + ∥ y 2 ∥ 2 2 − ∥ A G T y 2 ∥ 2 2 − ( n − 1 ) y 3 2 . (10) Let us choose y = ( y 1 , y 2 , y 3 ) y=(y_1,y_2,y_3) y = ( y 1 , y 2 , y 3 ) y 1 = 0 m y_1=0_m y 1 = 0 m y 3 = 1 y_3=1 y 3 = 1 y 2 ∈ { − 1 , 1 } n y_2\in\{-1,1\}^n y 2 ∈ { − 1 , 1 } n maxcut ( G ) \operatorname{maxcut}(G) maxcut ( G ) x = ( − A G y 2 , y 2 − 1 n , 1 ) x=(-A_Gy_2,y_2-\mathbf{1}_n,1) x = ( − A G y 2 , y 2 − 1 n , 1 ) x T M G x = 1 − 4 maxcut ( G ) x^{\mathsf T}M_Gx=1-4\operatorname{maxcut}(G) x T M G x = 1 − 4 maxcut ( G ) x ( M G x ) x(M_Gx) x ( M G x ) κ ^ ( M G ) ≥ κ ^ ( M G , x ) = maxcut ( G ) − 1 / 4 \hat{\kappa}(M_G)\ge\hat{\kappa}(M_G,x)=\operatorname{maxcut}(G)-1/4 κ ^ ( M G ) ≥ κ ^ ( M G , x ) = maxcut ( G ) − 1/4 (5) .
The upper bound κ ^ ( M G ) ≤ maxcut ( G ) − 1 / 4 \hat{\kappa}(M_G)\le\operatorname{maxcut}(G)-1/4 κ ^ ( M G ) ≤ maxcut ( G ) − 1/4 x ∈ R m + n + 1 x\in\mathbb{R}^{m+n+1} x ∈ R m + n + 1 ∑ i ∈ T M G x i ( M G x ) i \sum_{i\in T_{M_G}}x_i(M_Gx)_i ∑ i ∈ T M G x i ( M G x ) i (10) , noting that we can bound ∥ y 2 ∥ 2 2 − ∥ A G T y 2 ∥ 2 2 \lVert y_2\rVert_2^2-\lVert A_G^{\mathsf T}y_2\rVert_2^2 ∥ y 2 ∥ 2 2 − ∥ A G T y 2 ∥ 2 2 Proposition 7.12 .
Path following algorithm for P ∗ ( κ ) \mathcal{P}_*(\kappa) P ∗ ( κ )
The algorithm
Table 1.
We now describe a generic version of Newton’s method for solving LCPs with P ∗ ( κ ) \mathcal{P}_*(\kappa) P ∗ ( κ ) ( x , s ) ∈ F > 0 (x,s) \in\mathcal{F}_{>0} ( x , s ) ∈ F > 0 μ > 0 \mu> 0 μ > 0 ( x , s ) (x,s) ( x , s ) x s = μ 1 n xs=\mu1_n x s = μ 1 n
Definition 4.1 (Newton matrix and Newton step). Let ( x , s ) ∈ F > 0 (x,s) \in\mathcal{F}_{>0} ( x , s ) ∈ F > 0 μ > 0 \mu> 0 μ > 0 Newton matrix at ( x , s , μ ) (x,s,\mu) ( x , s , μ )
H x , μ ≔ ( I n + 1 μ X M X ) − 1 , H_{x,\mu} \coloneqq\left(I_n+\frac{1}{\mu}XMX\right)^{-1}, H x , μ : = ( I n + μ 1 X M X ) − 1 , where X = diag ( x ) X=\operatorname{diag}(x) X = diag ( x ) Newton step from ( x , s , μ ) (x,s,\mu) ( x , s , μ ) is ( Δ x , Δ s ) ∈ R 2 n (\Delta x,\Delta s) \in\mathbb{R}^{2n} ( Δ x , Δ s ) ∈ R 2 n
Δ x x = H x , μ err ( x , s , μ ) , Δ s = M Δ x . (11) \frac{\Delta x}{x}=H_{x,\mu}\operatorname{err}(x,s,\mu),\qquad\Delta s=M\Delta x. \tag*{(11)} x Δ x = H x , μ err ( x , s , μ ) , Δ s = M Δ x . (11) Corollary 2.5 guarantees that the Newton step is well defined for M ∈ P ∗ M \in\mathcal{P}_* M ∈ P ∗ β \beta β δ \delta δ N 2 ( β ) \mathcal{N}^2(\beta) N 2 ( β ) μ \mu μ 1 + δ 1+\delta 1 + δ κ \kappa κ § 8 .
Simple analysis with linear dependence on 1 + κ 1+\kappa 1 + κ
In this section, we give a simple analysis with linear dependence on 1 + κ 1+\kappa 1 + κ
Theorem 4.2. If M ∈ P ∗ ( κ ) M \in\mathcal{P}_*(\kappa) M ∈ P ∗ ( κ ) β ∈ [ 1 8 ( 1 + κ ) , 1 4 ( 1 + κ ) ] \beta\in\left[\frac{1}{8(1+\kappa)},\frac{1}{4(1+\kappa)}\right] β ∈ [ 8 ( 1 + κ ) 1 , 4 ( 1 + κ ) 1 ] δ ∈ [ 1 32 n ( 1 + κ ) , 1 16 n ( 1 + κ ) ] \delta\in\left[\frac{1}{32\sqrt{n}(1+\kappa)},\frac{1}{16\sqrt{n}(1+\kappa)}\right] δ ∈ [ 32 n ( 1 + κ ) 1 , 16 n ( 1 + κ ) 1 ] ( x 0 , s 0 , μ 0 ) ∈ N 2 ( β ) (x^0,s^0,\mu_0) \in\mathcal{N}^2(\beta) ( x 0 , s 0 , μ 0 ) ∈ N 2 ( β ) ( x o u t , s o u t , μ o u t ) ∈ N 2 ( β ) (x^{\mathrm{out}},s^{\mathrm{out}},\mu^{\mathrm{out}}) \in\mathcal{N}^2(\beta) ( x out , s out , μ out ) ∈ N 2 ( β ) μ o u t ≤ μ 1 \mu^{\mathrm{out}} \leq\mu_1 μ out ≤ μ 1 O ( n ( 1 + κ ) log ( μ 0 / μ 1 ) ) O\left(\sqrt{n}(1+\kappa)\log(\mu_0/\mu_1)\right) O ( n ( 1 + κ ) log ( μ 0 / μ 1 ) )
The following important identity will also be used in the later analysis:
Lemma 4.3 (Reducing centrality error). Let M ∈ P ∗ M \in\mathcal{P}_* M ∈ P ∗ ( x , s ) ∈ F > 0 (x,s) \in\mathcal{F}_{>0} ( x , s ) ∈ F > 0 μ > 0 \mu> 0 μ > 0 ( Δ x , Δ s ) (\Delta x,\Delta s) ( Δ x , Δ s ) (11) , and ( x ′ , s ′ ) = ( x + Δ x , s + Δ s ) (x',s')=(x+\Delta x,s+\Delta s) ( x ′ , s ′ ) = ( x + Δ x , s + Δ s )
err ( x ′ , s ′ , μ ) = ( H x , μ err ( x , s , μ ) ) 2 . \operatorname{err}(x',s',\mu)=\left(H_{x,\mu}\operatorname{err}(x,s,\mu)\right)^2. err ( x ′ , s ′ , μ ) = ( H x , μ err ( x , s , μ ) ) 2 . Proof. It follows from (11) that
err ( x ′ , s ′ , μ ) = 1 n − ( x + Δ x ) ( s + Δ s ) μ = 1 n − ( 1 n + Δ x x ) ( x s + X M X Δ x x ) μ = 1 n − ( 1 n + Δ x x ) ( ( I n + 1 μ X M X ) Δ x x + ( x s μ − Δ x x ) ) = 1 n − ( 1 n + Δ x x ) ( 1 n − x s μ + x s μ − Δ x x ) = ( Δ x x ) 2 = ( H x , μ err ( x , s , μ ) ) 2 . (12) \begin{aligned}
\operatorname{err}(x',s',\mu)
&=1_n-\frac{(x+\Delta x)(s+\Delta s)}{\mu}
=1_n-\frac{\left(1_n+\frac{\Delta x}{x}\right)\left(xs+XMX\frac{\Delta x}{x}\right)}{\mu}\\
&=1_n-\left(1_n+\frac{\Delta x}{x}\right)\left(\left(I_n+\frac{1}{\mu}XMX\right)\frac{\Delta x}{x}+\left(\frac{xs}{\mu}-\frac{\Delta x}{x}\right)\right)\\
&=1_n-\left(1_n+\frac{\Delta x}{x}\right)\left(1_n-\frac{xs}{\mu}+\frac{xs}{\mu}-\frac{\Delta x}{x}\right)\\
&=\left(\frac{\Delta x}{x}\right)^2
=\left(H_{x,\mu}\operatorname{err}(x,s,\mu)\right)^2. \tag*{(12)}
\end{aligned} err ( x ′ , s ′ , μ ) = 1 n − μ ( x + Δ x ) ( s + Δ s ) = 1 n − μ ( 1 n + x Δ x ) ( x s + X M X x Δ x ) = 1 n − ( 1 n + x Δ x ) ( ( I n + μ 1 X M X ) x Δ x + ( μ x s − x Δ x ) ) = 1 n − ( 1 n + x Δ x ) ( 1 n − μ x s + μ x s − x Δ x ) = ( x Δ x ) 2 = ( H x , μ err ( x , s , μ ) ) 2 . (12) The proof of Theorem 4.2 follows immediately from the following lemma:
Lemma 4.4 (Progress lemma). Assume M ∈ P ∗ ( κ ) M \in\mathcal{P}_*(\kappa) M ∈ P ∗ ( κ ) For any β ∈ ( 0 , 1 4 ( 1 + κ ) ] \beta\in\left(0,\frac{1}{4(1+\kappa)}\right] β ∈ ( 0 , 4 ( 1 + κ ) 1 ] δ ∈ ( 0 , β 2 n ] \delta\in\left(0,\frac{\beta}{2\sqrt{n}}\right] δ ∈ ( 0 , 2 n β ] ( x , s , μ ) ∈ N 2 ( β ) (x,s,\mu) \in\mathcal{N}^2(\beta) ( x , s , μ ) ∈ N 2 ( β ) μ ′ = μ / ( 1 + δ ) \mu'=\mu/(1+\delta) μ ′ = μ / ( 1 + δ ) ( Δ x , Δ s ) (\Delta x,\Delta s) ( Δ x , Δ s ) (11) for μ ′ \mu' μ ′ ( x ′ , s ′ ) = ( x + Δ x , s + Δ s ) (x',s')=(x+\Delta x,s+\Delta s) ( x ′ , s ′ ) = ( x + Δ x , s + Δ s ) ( x ′ , s ′ , μ ′ ) ∈ N 2 ( β ) (x',s',\mu') \in\mathcal{N}^2(\beta) ( x ′ , s ′ , μ ′ ) ∈ N 2 ( β )
Proof. We start by bounding err ( x , s , μ ′ ) \operatorname{err}(x,s,\mu') err ( x , s , μ ′ ) ∥ x s / μ − 1 n ∥ ≤ β \lVert xs/\mu-1_n\rVert\leq\beta ∥ x s / μ − 1 n ∥ ≤ β ( x , s , μ ) ∈ N 2 ( β ) (x,s,\mu) \in\mathcal{N}^2(\beta) ( x , s , μ ) ∈ N 2 ( β ) δ ≤ β 2 n \delta\leq\frac{\beta}{2\sqrt{n}} δ ≤ 2 n β
∥ err ( x , s , μ ′ ) ∥ 2 = ∥ x s μ ′ − 1 n ∥ 2 = ∥ x s μ − 1 n ∥ 2 + δ ∥ x s μ ∥ 2 ≤ β + β 2 n ⋅ ( 1 + β ) n ≤ 2 β . \begin{aligned}
\lVert\operatorname{err}(x,s,\mu')\rVert_2
&=\left\lVert\frac{xs}{\mu'}-1_n\right\rVert_2
=\left\lVert\frac{xs}{\mu}-1_n\right\rVert_2+\delta\left\lVert\frac{xs}{\mu}\right\rVert_2\\
&\leq\beta+\frac{\beta}{2\sqrt{n}}\cdot(1+\beta)\sqrt{n}\leq2\beta.
\end{aligned} ∥ err ( x , s , μ ′ ) ∥ 2 = μ ′ x s − 1 n 2 = μ x s − 1 n 2 + δ μ x s 2 ≤ β + 2 n β ⋅ ( 1 + β ) n ≤ 2 β . From Lemma 4.3, we have err ( x ′ , s ′ , μ ′ ) = ( H x , μ ′ err ( x , s , μ ′ ) ) 2 \operatorname{err}(x',s',\mu')=\left(H_{x,\mu'}\operatorname{err}(x,s,\mu')\right)^2 err ( x ′ , s ′ , μ ′ ) = ( H x , μ ′ err ( x , s , μ ′ ) ) 2 κ ^ ( M ) ≤ κ \hat{\kappa}(M)\leq\kappa κ ^ ( M ) ≤ κ ∥ H x , μ ′ ∥ 2 ≤ 1 + κ \lVert H_{x,\mu'}\rVert_2\leq\sqrt{1+\kappa} ∥ H x , μ ′ ∥ 2 ≤ 1 + κ
∥ err ( x ′ , s ′ , μ ′ ) ∥ 2 ≤ ∥ H x , μ ′ err ( x , s , μ ′ ) ∥ 2 2 ≤ ( 1 + κ ) ∥ err ( x , s , μ ′ ) ∥ 2 2 ≤ ( 1 + κ ) 4 β 2 ≤ β , \begin{aligned}
\lVert\operatorname{err}(x',s',\mu')\rVert_2
&\leq\lVert H_{x,\mu'}\operatorname{err}(x,s,\mu')\rVert_2^2
\leq(1+\kappa)\lVert\operatorname{err}(x,s,\mu')\rVert_2^2\\
&\leq(1+\kappa)4\beta^2\leq\beta,
\end{aligned} ∥ err ( x ′ , s ′ , μ ′ ) ∥ 2 ≤ ∥ H x , μ ′ err ( x , s , μ ′ ) ∥ 2 2 ≤ ( 1 + κ ) ∥ err ( x , s , μ ′ ) ∥ 2 2 ≤ ( 1 + κ ) 4 β 2 ≤ β , and by the bound on β \beta β ( x ′ , s ′ ) ∈ F > 0 (x',s')\in\mathcal{F}_{>0} ( x ′ , s ′ ) ∈ F > 0 ( x ′ , s ′ , μ ′ ) ∈ N 2 ( β ) (x',s',\mu')\in\mathcal{N}^2(\beta) ( x ′ , s ′ , μ ′ ) ∈ N 2 ( β )
Proof of Theorem 4.2. Starting from ( x 0 , s 0 , μ 0 ) ∈ N 2 ( β ) (x^0,s^0,\mu_0) \in\mathcal{N}^2(\beta) ( x 0 , s 0 , μ 0 ) ∈ N 2 ( β ) ( x , s , μ ) ∈ N 2 ( β ) (x,s,\mu) \in\mathcal{N}^2(\beta) ( x , s , μ ) ∈ N 2 ( β ) μ ≤ μ 0 \mu\le\mu_0 μ ≤ μ 0 ( Δ x , Δ s ) (\Delta x,\Delta s) ( Δ x , Δ s ) ( x , s ) (x,s) ( x , s ) μ ′ = μ / ( 1 + δ ) \mu'=\mu/(1+\delta) μ ′ = μ / ( 1 + δ ) (11) . With β ∈ [ 1 8 ( 1 + κ ) , 1 4 ( 1 + κ ) ] \beta\in\left[\frac{1}{8(1+\kappa)},\frac{1}{4(1+\kappa)}\right] β ∈ [ 8 ( 1 + κ ) 1 , 4 ( 1 + κ ) 1 ] δ ∈ [ 1 32 n ( 1 + κ ) , 1 16 n ( 1 + κ ) ] \delta\in\left[\frac{1}{32\sqrt{n}(1+\kappa)},\frac{1}{16\sqrt{n}(1+\kappa)}\right] δ ∈ [ 32 n ( 1 + κ ) 1 , 16 n ( 1 + κ ) 1 ] δ ≤ β 2 n \delta\le\frac{\beta}{2\sqrt{n}} δ ≤ 2 n β ( x + Δ x , s + Δ s , μ ′ ) ∈ N 2 ( β ) (x+\Delta x,s+\Delta s,\mu') \in\mathcal{N}^2(\beta) ( x + Δ x , s + Δ s , μ ′ ) ∈ N 2 ( β ) Lemma 4.4 . Let ( x + Δ x , s + Δ s ) (x+\Delta x,s+\Delta s) ( x + Δ x , s + Δ s ) μ ′ \mu' μ ′ δ \delta δ
μ ′ ≤ ( 1 − 1 64 n ( 1 + κ ) ) μ . \mu' \le\left(1-\frac{1}{64\sqrt{n}(1+\kappa)}\right)\mu. μ ′ ≤ ( 1 − 64 n ( 1 + κ ) 1 ) μ . Thus, it takes O ( n ( 1 + κ ) log ( μ 0 / μ 1 ) ) O\left(\sqrt{n}(1+\kappa)\log(\mu_0/\mu_1)\right) O ( n ( 1 + κ ) log ( μ 0 / μ 1 ) ) ( x o u t , s o u t , μ o u t ) ∈ N 2 ( β ) (x^{\mathrm{out}},s^{\mathrm{out}},\mu^{\mathrm{out}}) \in\mathcal{N}^2(\beta) ( x out , s out , μ out ) ∈ N 2 ( β ) μ o u t ≤ μ 1 \mu^{\mathrm{out}} \le\mu_1 μ out ≤ μ 1
An improved analysis via Rump weights
In what follows we show that the standard IPM based on Newton steps actually can reduce μ \mu μ O ( n 5 / 4 1 + κ ) O(n^{5/4}\sqrt{1+\kappa}) O ( n 5/4 1 + κ ) Theorem 2.10 .
Theorem 5.1. Let M ∈ P ⋆ ( κ ) M \in\mathcal{P}_\star(\kappa) M ∈ P ⋆ ( κ ) β ∈ [ 1 20480 n 3 / 2 , 1 16384 n 3 / 2 ] \beta\in\left[\frac{1}{20480n^{3/2}},\frac{1}{16384n^{3/2}}\right] β ∈ [ 20480 n 3/2 1 , 16384 n 3/2 1 ] δ ∈ [ 1 9160 n 5 / 4 1 + κ , 1 4580 n 5 / 4 1 + κ ] \delta\in\left[\frac{1}{9160n^{5/4}\sqrt{1+\kappa}},\frac{1}{4580n^{5/4}\sqrt{1+\kappa}}\right] δ ∈ [ 9160 n 5/4 1 + κ 1 , 4580 n 5/4 1 + κ 1 ] ( x 0 , s 0 , μ 0 ) ∈ N R ( β ) (x^0,s^0,\mu_0) \in\mathcal{N}^{R}(\beta) ( x 0 , s 0 , μ 0 ) ∈ N R ( β ) ( x o u t , s o u t , μ o u t ) ∈ N R ( β ) (x^{\mathrm{out}},s^{\mathrm{out}},\mu^{\mathrm{out}}) \in\mathcal{N}^{R}(\beta) ( x out , s out , μ out ) ∈ N R ( β ) μ o u t ≤ μ 1 \mu^{\mathrm{out}} \le\mu_1 μ out ≤ μ 1 O ( n 5 / 4 1 + κ ⋅ log ( μ 0 / μ 1 ) ) O\left(n^{5/4}\sqrt{1+\kappa}\cdot\log(\mu_0/\mu_1)\right) O ( n 5/4 1 + κ ⋅ log ( μ 0 / μ 1 ) )
To achieve this result, we need to show that Newton steps converge in a local metric, defined by Rump weights, which we formally define in what follows.
The Rump weights neighborhood
For the proof of Theorem 5.1, we introduce the following weights based on Rump’s theorem (Theorem 2.10). Throughout, we set the parameter K : = 8 K := 8 K := 8 K ≥ 5 + 2 2 K \ge5+2\sqrt{2} K ≥ 5 + 2 2
Definition 5.2. Let H = ( I n + A ) − 1 H=(I_n+A)^{-1} H = ( I n + A ) − 1 A ∈ P ⋆ A \in\mathcal{P}_\star A ∈ P ⋆ q H ( i ) : = π i q_H(i):=\pi_i q H ( i ) := π i (6) for the weights ρ ( i , j ) : = ∣ H i j ∣ / K \rho(i,j):=|H_{ij}|/K ρ ( i , j ) := ∣ H ij ∣/ K Q H : = diag ( q H ) Q_H:=\operatorname{diag}(q_H) Q H := diag ( q H ) j ∈ [ n ] j \in[n] j ∈ [ n ]
q H ( j ) : = max { ∏ k = 1 ℓ ∣ H i k i k + 1 ∣ K : i 1 , … , i ℓ + 1 = j is a simple path } , q_H(j):=\max\left\{\prod_{k=1}^{\ell}\frac{|H_{i_k i_{k+1}}|}{K}: i_1,\ldots,i_{\ell+1}=j\text{ is a simple path}\right\}, q H ( j ) := max { k = 1 ∏ ℓ K ∣ H i k i k + 1 ∣ : i 1 , … , i ℓ + 1 = j is a simple path } , with the convention that the product is 1 for an empty path.
By applying Corollary 2.12 to the matrix H = ( I n + A ) − 1 H=(I_n+A)^{-1} H = ( I n + A ) − 1 H H H O ( 1 ) O(1) O ( 1 ) H H H
Proposition 5.3. Let A ∈ P ⋆ ∩ R n × n A \in\mathcal{P}_\star\cap\mathbb{R}^{n\times n} A ∈ P ⋆ ∩ R n × n H = ( I + A ) − 1 H=(I+A)^{-1} H = ( I + A ) − 1 i ∈ [ n ] i \in[n] i ∈ [ n ] H i i ∈ ( 0 , 1 ] H_{ii} \in(0,1] H ii ∈ ( 0 , 1 ]
Proof. Since A ∈ P ⋆ A \in\mathcal{P}_\star A ∈ P ⋆ I + A I+A I + A Corollary 2.5 . Let i ∈ [ n ] i \in[n] i ∈ [ n ] h = H ∙ i = H e i h=H_{\mathbin{\bullet}i}=He_i h = H ∙ i = H e i
A h = A ( I + A ) − 1 e i = e i − ( I + A ) − 1 e i = e i − h . Ah=A(I+A)^{-1}e_i=e_i-(I+A)^{-1}e_i=e_i-h. A h = A ( I + A ) − 1 e i = e i − ( I + A ) − 1 e i = e i − h . Hence, h i ( A h ) i = h i ( 1 − h i ) h_i(Ah)_i=h_i(1-h_i) h i ( A h ) i = h i ( 1 − h i ) h j ( A h ) j = − h j 2 ≤ 0 h_j(Ah)_j=-h_j^2\le0 h j ( A h ) j = − h j 2 ≤ 0 j ≠ i j\ne i j = i A A A h ∘ A h ≤ 0 n h\circ Ah\le0_n h ∘ A h ≤ 0 n h ∘ A h = 0 n h\circ Ah=0_n h ∘ A h = 0 n h i ∉ ( 0 , 1 ] h_i\notin(0,1] h i ∈ / ( 0 , 1 ] A A A h = H ∙ i = 0 n h=H_{\mathbin{\bullet}i}=0_n h = H ∙ i = 0 n H H H
Lemma 5.4. For A ∈ P ⋆ ∩ R n × n A \in\mathcal{P}_\star\cap\mathbb{R}^{n\times n} A ∈ P ⋆ ∩ R n × n H = ( I n + A ) − 1 H=(I_n+A)^{-1} H = ( I n + A ) − 1 ζ ( H ) < 6 \zeta(H)<6 ζ ( H ) < 6 q H q_H q H ∥ Q H H Q H − 1 ∥ max ≤ K \lVert Q_H H Q_H^{-1}\rVert_{\max}\le K ∥ Q H H Q H − 1 ∥ m a x ≤ K
Proof. By Lemma 2.8 , ρ 0 S ( H ) ≤ 1 \rho^S_0(H)\le1 ρ 0 S ( H ) ≤ 1 Theorem 2.10 , ζ ( H ) ≤ ( 3 + 2 2 ) ρ 0 S ( H ) ≤ 3 + 2 2 < 6 \zeta(H)\le(3+2\sqrt{2})\rho^S_0(H)\le3+2\sqrt{2}<6 ζ ( H ) ≤ ( 3 + 2 2 ) ρ 0 S ( H ) ≤ 3 + 2 2 < 6 i ∈ [ n ] i\in[n] i ∈ [ n ] q H ( i ) q_H(i) q H ( i ) i ∈ [ n ] i\in[n] i ∈ [ n ] ∣ Q H H Q H − 1 ∣ i i = ∣ H i i ∣ ≤ 1 \left|Q_HHQ_H^{-1}\right|_{ii}=|H_{ii}|\le1 Q H H Q H − 1 ii = ∣ H ii ∣ ≤ 1 Proposition 5.3 . Thus as a result of Corollary 2.12, we get ∥ Q H H Q H − 1 ∥ max ≤ K \lVert Q_HHQ_H^{-1}\rVert_{\max}\le K ∥ Q H H Q H − 1 ∥ m a x ≤ K
Note that for M ∈ P ⋆ M \in\mathcal{P}_{\star} M ∈ P ⋆ x ∈ R > 0 n x \in\mathbb{R}^{n}_{>0} x ∈ R > 0 n μ > 0 \mu> 0 μ > 0 H x , μ H_{x,\mu} H x , μ H x , μ = ( I n + 1 μ X M X ) − 1 H_{x,\mu} = (I_n + \frac{1}{\mu}XMX)^{-1} H x , μ = ( I n + μ 1 X M X ) − 1 X = diag ( x ) X = \operatorname{diag}(x) X = diag ( x ) 1 μ X M X \frac{1}{\mu}XMX μ 1 X M X M M M H = H x , μ H = H_{x,\mu} H = H x , μ
∥ Q H x , μ v ∥ 2 = ⟨ q H x , μ 2 , v 2 ⟩ . \left\|Q_{H_{x,\mu}}v\right\|_2 = \sqrt{\left\langle q_{H_{x,\mu}}^2,v^2\right\rangle}. Q H x , μ v 2 = ⟨ q H x , μ 2 , v 2 ⟩ . Definition 5.5. Consider an instance of LCP ( M , q ) \operatorname{LCP}(M,q) LCP ( M , q ) M ∈ P ⋆ M \in\mathcal{P}_{\star} M ∈ P ⋆ β ≥ 0 \beta\ge0 β ≥ 0 Rump-norm neighborhood of the central path is defined as
N R ( β ) : = { ( x , s , μ ) : ( x , s ) ∈ F > 0 , μ ∈ R > 0 , ∥ Q H x , μ err ( x , s , μ ) ∥ 2 ≤ β } . \mathcal{N}^{R}(\beta) := \left\{(x,s,\mu) : (x,s) \in\mathcal{F}_{>0},\ \mu\in\mathbb{R}_{>0},\ \left\|Q_{H_{x,\mu}}\operatorname{err}(x,s,\mu)\right\|_2 \le\beta\right\}. N R ( β ) := { ( x , s , μ ) : ( x , s ) ∈ F > 0 , μ ∈ R > 0 , Q H x , μ err ( x , s , μ ) 2 ≤ β } . where H x , μ H_{x,\mu} H x , μ err ( x , s , μ ) \operatorname{err}(x,s,\mu) err ( x , s , μ ) (4) .
In Section 5.3, we prove the following bound on the q H q_H q H
Lemma 5.6 (Bounding Rump weights). For A ∈ P ⋆ ( κ ) A \in\mathcal{P}_{\star}(\kappa) A ∈ P ⋆ ( κ ) H = ( I n + A ) − 1 H = (I_n + A)^{-1} H = ( I n + A ) − 1 Q H = diag ( q H ) Q_H = \operatorname{diag}(q_H) Q H = diag ( q H ) ∥ Q H H v ∥ ∞ ≤ κ 2 1 + κ ∥ v ∥ 2 \left\|Q_HHv\right\|_{\infty} \le\frac{\kappa}{2}\sqrt{1+\kappa}\left\|v\right\|_2 ∥ Q H H v ∥ ∞ ≤ 2 κ 1 + κ ∥ v ∥ 2 v ∈ R n v \in\mathbb{R}^n v ∈ R n 1 n ≤ q H ≤ κ 2 1 + κ ⋅ 1 n \mathbf{1}_n \le q_H \le\frac{\kappa}{2}\sqrt{1+\kappa}\cdot\mathbf{1}_n 1 n ≤ q H ≤ 2 κ 1 + κ ⋅ 1 n
Note that because of q H ≥ 1 n q_H \ge\mathbf{1}_n q H ≥ 1 n N R ( β ) ⊆ N 2 ( β ) \mathcal{N}^{R}(\beta) \subseteq\mathcal{N}^{2}(\beta) N R ( β ) ⊆ N 2 ( β )
N R ( β ) ⊆ N 2 ( β ) ⊆ N R ( κ 2 1 + κ ⋅ β ) , ∀ κ ≥ κ ^ ( M ) . (13) \mathcal{N}^{R}(\beta) \subseteq\mathcal{N}^{2}(\beta) \subseteq\mathcal{N}^{R}\left(\frac{\kappa}{2}\sqrt{1+\kappa}\cdot\beta\right), \qquad\forall\ \kappa\ge\widehat{\kappa}(M). \tag*{(13)} N R ( β ) ⊆ N 2 ( β ) ⊆ N R ( 2 κ 1 + κ ⋅ β ) , ∀ κ ≥ κ ( M ) . (13) We need one more bound on the stability of Rump weights, proven in Section 5.4.
Lemma 5.7 (Stability of Rump weights). For A ∈ P ⋆ ∩ R n × n A \in\mathcal{P}_{\star} \cap\mathbb{R}^{n \times n} A ∈ P ⋆ ∩ R n × n u ∈ R > 0 n u \in\mathbb{R}^{n}_{>0} u ∈ R > 0 n U = diag ( u ) U = \operatorname{diag}(u) U = diag ( u ) H : = ( I n + A ) − 1 H := (I_n + A)^{-1} H := ( I n + A ) − 1 H ′ : = ( I n + U A U ) − 1 H' := (I_n + UAU)^{-1} H ′ := ( I n + U A U ) − 1 q H , q H ′ q_H,q_{H'} q H , q H ′
∥ log q H ′ − log q H ∥ ∞ ≤ 16 K ∥ log ( u ) ∥ 1 . (14) \left\|\log q_{H'}-\log q_H\right\|_{\infty} \le16K\left\|\log(u)\right\|_1. \tag*{(14)} ∥ log q H ′ − log q H ∥ ∞ ≤ 16 K ∥ log ( u ) ∥ 1 . (14) Analyzing Newton steps in the Rump neighborhood
Using the properties provided in this section, we can show that Newton’s method converges in the Rump neighborhood at a rate independent of the handicap number of the input matrix. Therefore, assuming we start with a point that is in a small Rump neighborhood of the central path, running Newton’s method for O ( log log κ + log log n ) O(\log\log\kappa+\log\log n) O ( log log κ + log log n ) O ( n − 3 / 2 ) O(n^{-3/2}) O ( n − 3/2 )
Proposition 5.8 (Reducing centrality error). Let M ∈ P ⋆ ( κ ) M \in\mathcal{P}_{\star}(\kappa) M ∈ P ⋆ ( κ ) ( x , s , μ ) ∈ N R ( β ) (x,s,\mu) \in\mathcal{N}^{R}(\beta) ( x , s , μ ) ∈ N R ( β ) β ∈ ( 0 , 1 64 K 2 n 3 / 2 ] \beta\in\left(0,\frac{1}{64K^2n^{3/2}}\right] β ∈ ( 0 , 64 K 2 n 3/2 1 ] ( Δ x , Δ s ) (\Delta x,\Delta s) ( Δ x , Δ s ) (11) , and ( x ′ , s ′ ) = ( x + Δ x , s + Δ s ) (x',s')=(x+\Delta x,s+\Delta s) ( x ′ , s ′ ) = ( x + Δ x , s + Δ s ) ( x ′ , s ′ , μ ) ∈ N R ( 2 K 2 n 3 / 2 β 2 ) (x',s',\mu) \in\mathcal{N}^{R}(2K^2n^{3/2}\beta^2) ( x ′ , s ′ , μ ) ∈ N R ( 2 K 2 n 3/2 β 2 )
The proof requires the following simple technical lemma that we will also use later:
Lemma 5.9. For a matrix H = ( I n + A ) − 1 H=(I_n+A)^{-1} H = ( I n + A ) − 1 q H q_H q H v ∈ R n v \in\mathbb{R}^n v ∈ R n
∥ Q H H v ∥ ∞ ≤ K n 1 / 2 ⋅ ∥ Q H v ∥ 2 , and ∥ Q H ( H v ) 2 ∥ 2 ≤ K 2 n 3 / 2 ∥ Q H v ∥ 2 2 . \left\|Q_HHv\right\|_{\infty} \le K n^{1/2}\cdot\left\|Q_Hv\right\|_2,\qquad\text{and}\qquad\left\|Q_H(Hv)^2\right\|_2 \le K^2n^{3/2}\left\|Q_Hv\right\|_2^2. ∥ Q H H v ∥ ∞ ≤ K n 1/2 ⋅ ∥ Q H v ∥ 2 , and Q H ( H v ) 2 2 ≤ K 2 n 3/2 ∥ Q H v ∥ 2 2 . Proof. Recall that q H ≥ 1 n q_H \ge\mathbf{1}_n q H ≥ 1 n ∥ H v ∥ ∞ ≤ ∥ Q H H v ∥ ∞ \left\|Hv\right\|_{\infty} \le\left\|Q_HHv\right\|_{\infty} ∥ H v ∥ ∞ ≤ ∥ Q H H v ∥ ∞ ∥ Q H H Q H − 1 ∥ max ≤ K \left\|Q_HHQ_H^{-1}\right\|_{\max} \le K Q H H Q H − 1 m a x ≤ K ∥ Q H H Q H − 1 ∥ 2 → ∞ ≤ K n 1 / 2 \left\|Q_HHQ_H^{-1}\right\|_{2\to\infty} \le Kn^{1/2} Q H H Q H − 1 2 → ∞ ≤ K n 1/2
∥ Q H H v ∥ ∞ ≤ ∥ Q H H Q H − 1 ∥ 2 → ∞ ⋅ ∥ Q H v ∥ 2 ≤ K n 1 / 2 ⋅ ∥ Q H v ∥ 2 . \left\|Q_HHv\right\|_{\infty} \le\left\|Q_HHQ_H^{-1}\right\|_{2\to\infty}\cdot\left\|Q_Hv\right\|_2 \le Kn^{1/2}\cdot\left\|Q_Hv\right\|_2. ∥ Q H H v ∥ ∞ ≤ Q H H Q H − 1 2 → ∞ ⋅ ∥ Q H v ∥ 2 ≤ K n 1/2 ⋅ ∥ Q H v ∥ 2 . The second claim then follows by
∥ Q H ( H v ) 2 ∥ 2 ≤ ∥ H v ∥ ∞ ∥ Q H H v ∥ 2 ≤ ∥ Q H H v ∥ ∞ ∥ Q H H v ∥ 2 ≤ n 1 / 2 ∥ Q H H v ∥ ∞ 2 . \left\|Q_H(Hv)^2\right\|_2 \le\left\|Hv\right\|_{\infty}\left\|Q_HHv\right\|_2 \le\left\|Q_HHv\right\|_{\infty}\left\|Q_HHv\right\|_2 \le n^{1/2}\left\|Q_HHv\right\|_{\infty}^2. Q H ( H v ) 2 2 ≤ ∥ H v ∥ ∞ ∥ Q H H v ∥ 2 ≤ ∥ Q H H v ∥ ∞ ∥ Q H H v ∥ 2 ≤ n 1/2 ∥ Q H H v ∥ ∞ 2 . Proof of Proposition 5.8. Let us simplify the notation by setting H : = H x , μ H := H_{x,\mu} H := H x , μ H ′ = H x ′ , μ H' = H_{x',\mu} H ′ = H x ′ , μ err : = err ( x , s , μ ) \operatorname{err} := \operatorname{err}(x,s,\mu) err := err ( x , s , μ ) err ′ : = err ( x ′ , s ′ , μ ) \operatorname{err}' := \operatorname{err}(x',s',\mu) err ′ := err ( x ′ , s ′ , μ ) (11) , x ′ = x ( 1 n + H err ) x' = x(\mathbf{1}_n + H\operatorname{err}) x ′ = x ( 1 n + H err ) ∥ Q H H err ∥ 2 ≤ β \lVert Q_H H\operatorname{err}\rVert_2 \le\beta ∥ Q H H err ∥ 2 ≤ β ∥ Q H ′ err ′ ∥ 2 ≤ 2 K 2 n 3 / 2 β 2 \lVert Q_{H'}\operatorname{err}'\rVert_2 \le2K^2n^{3/2}\beta^2 ∥ Q H ′ err ′ ∥ 2 ≤ 2 K 2 n 3/2 β 2
First, let us bound the new centrality error err ′ \operatorname{err}' err ′ Q H Q_H Q H err ′ = ( H err ) 2 \operatorname{err}' = (H\operatorname{err})^2 err ′ = ( H err ) 2
∥ Q H err ′ ∥ 2 = ∥ Q H ( H err ) 2 ∥ 2 ≤ K 2 n 3 / 2 ∥ Q H err ∥ 2 2 ≤ K 2 n 3 / 2 β 2 . (15) \lVert Q_H\operatorname{err}'\rVert_2 = \lVert Q_H(H\operatorname{err})^2\rVert_2 \le K^2n^{3/2}\lVert Q_H\operatorname{err}\rVert_2^2 \le K^2n^{3/2}\beta^2. \tag*{(15)} ∥ Q H err ′ ∥ 2 = ∥ Q H ( H err ) 2 ∥ 2 ≤ K 2 n 3/2 ∥ Q H err ∥ 2 2 ≤ K 2 n 3/2 β 2 . (15) We finish the proof by showing that when performing a Newton step in this neighborhood, Rump weights increase only by at most a constant factor. Let u : = 1 n + H err u := \mathbf{1}_n + H\operatorname{err} u := 1 n + H err x x x x ′ x' x ′ ∣ log ( 1 + α ) ∣ ≤ 2 ∣ α ∣ \lvert\log(1+\alpha)\rvert\le2\lvert\alpha\rvert ∣ log ( 1 + α )∣ ≤ 2 ∣ α ∣ α ≥ − 1 / 2 \alpha\ge-1/2 α ≥ − 1/2
∥ log u ∥ ∞ ≤ 2 ∥ u − 1 n ∥ ∞ = 2 ∥ H err ∥ ∞ ≤ 2 ∥ Q H H err ∥ ∞ ≤ 2 K n 1 / 2 ∥ Q H err ∥ 2 ≤ 2 K n 1 / 2 β . (16) \lVert\log u\rVert_\infty\le2\lVert u-\mathbf{1}_n\rVert_\infty= 2\lVert H\operatorname{err}\rVert_\infty\le2\lVert Q_HH\operatorname{err}\rVert_\infty\le2Kn^{1/2}\lVert Q_H\operatorname{err}\rVert_2 \le2Kn^{1/2}\beta. \tag*{(16)} ∥ log u ∥ ∞ ≤ 2 ∥ u − 1 n ∥ ∞ = 2 ∥ H err ∥ ∞ ≤ 2 ∥ Q H H err ∥ ∞ ≤ 2 K n 1/2 ∥ Q H err ∥ 2 ≤ 2 K n 1/2 β . (16) Then, applying Lemma 5.7 to A : = 1 μ X M X A := \frac{1}{\mu}XMX A := μ 1 X M X H ′ = ( I n + U A U ) − 1 H' = (I_n + UAU)^{-1} H ′ = ( I n + U A U ) − 1 U = diag ( u ) U = \operatorname{diag}(u) U = diag ( u )
∥ log q H ′ − log q H ∥ ∞ ≤ 16 K ∥ log u ∥ 1 ≤ 16 K ⋅ n ∥ log u ∥ ∞ ≤ 32 K 2 n 3 / 2 β ≤ 1 2 , \lVert\log q_{H'}-\log q_H\rVert_\infty\le16K\lVert\log u\rVert_1 \le16K\cdot n\lVert\log u\rVert_\infty\le32K^2n^{3/2}\beta\le\frac{1}{2}, ∥ log q H ′ − log q H ∥ ∞ ≤ 16 K ∥ log u ∥ 1 ≤ 16 K ⋅ n ∥ log u ∥ ∞ ≤ 32 K 2 n 3/2 β ≤ 2 1 , by the range of β \beta β ∥ q H ′ / q H ∥ ∞ ≤ e 1 / 2 < 2 \lVert q_{H'}/q_H\rVert_\infty\le e^{1/2}<2 ∥ q H ′ / q H ∥ ∞ ≤ e 1/2 < 2 (15) , ∥ Q H ′ err ′ ∥ ≤ 2 ∥ Q H err ′ ∥ ≤ 2 K 2 n 3 / 2 β 2 \lVert Q_{H'}\operatorname{err}'\rVert\le2\lVert Q_H\operatorname{err}'\rVert\le2K^2n^{3/2}\beta^2 ∥ Q H ′ err ′ ∥ ≤ 2 ∥ Q H err ′ ∥ ≤ 2 K 2 n 3/2 β 2 □ \square □
Therefore, starting from a point in the O ( n − 3 / 2 ) O(n^{-3/2}) O ( n − 3/2 ) O ~ ( n 2 1 + κ ) \widetilde{O}(n^2\sqrt{1+\kappa}) O ( n 2 1 + κ ) μ \mu μ 1 + δ 1+\delta 1 + δ δ ∥ Q H 1 n ∥ = δ ∥ q H ∥ = δ ⋅ O ( n 1 / 2 1 + κ ) \delta\lVert Q_H\mathbf{1}_n\rVert= \delta\lVert q_H\rVert= \delta\cdot O(n^{1/2}\sqrt{1+\kappa}) δ ∥ Q H 1 n ∥ = δ ∥ q H ∥ = δ ⋅ O ( n 1/2 1 + κ ) δ = O ( 1 / ( n 2 1 + κ ) ) \delta= O(1/(n^2\sqrt{1+\kappa})) δ = O ( 1/ ( n 2 1 + κ ))
Proof of Theorem 5.1
The proof of Theorem 5.1 hinges on Lemma 5.10. Starting from a sufficiently small Rump norm neighborhood of radius O ( n − 3 / 2 ) O(n^{-3/2}) O ( n − 3/2 ) μ \mu μ 1 + δ 1+\delta 1 + δ δ = 1 / Ω ( n 5 / 4 1 + κ ) \delta= 1/\Omega(n^{5/4}\sqrt{1+\kappa}) δ = 1/Ω ( n 5/4 1 + κ ) μ \mu μ
Lemma 5.10 (Progress lemma). Let M ∈ P ⋆ ( κ ) M \in\mathcal{P}_\star(\kappa) M ∈ P ⋆ ( κ ) ( x , s , μ ) ∈ N R ( β ) (x,s,\mu) \in\mathcal{N}^{R}(\beta) ( x , s , μ ) ∈ N R ( β ) β ∈ ( 0 , 1 16384 n 3 / 2 ] \beta\in\left(0,\frac{1}{16384n^{3/2}}\right] β ∈ ( 0 , 16384 n 3/2 1 ] μ ′ = μ 1 + δ \mu' = \frac{\mu}{1+\delta} μ ′ = 1 + δ μ 0 < δ ≤ β 1024 n ( 1 + κ ) 0 < \delta\le\sqrt{\frac{\beta}{1024n(1+\kappa)}} 0 < δ ≤ 1024 n ( 1 + κ ) β ( Δ x , Δ s ) (\Delta x,\Delta s) ( Δ x , Δ s ) μ ′ \mu' μ ′
Δ x x = H x , μ ′ err ( x , s , μ ′ ) , Δ s = M Δ x , \frac{\Delta x}{x} = H_{x,\mu'}\operatorname{err}(x,s,\mu'),\qquad\Delta s = M\Delta x, x Δ x = H x , μ ′ err ( x , s , μ ′ ) , Δ s = M Δ x , and let ( x ′ , s ′ ) = ( x + Δ x , s + Δ s ) (x',s')=(x+\Delta x,s+\Delta s) ( x ′ , s ′ ) = ( x + Δ x , s + Δ s ) ( x ′ , s ′ , μ ′ ) ∈ N R ( β ) (x',s',\mu')\in\mathcal{N}^{R}(\beta) ( x ′ , s ′ , μ ′ ) ∈ N R ( β )
Proof. Since ( x , s , μ ) ∈ N R ( β ) (x,s,\mu)\in\mathcal{N}^{R}(\beta) ( x , s , μ ) ∈ N R ( β ) ∥ Q H x , μ err ( x , s , μ ) ∥ ≤ β \lVert Q_{H_{x,\mu}}\operatorname{err}(x,s,\mu)\rVert\le\beta ∥ Q H x , μ err ( x , s , μ )∥ ≤ β μ ′ = μ 1 + δ \mu'=\frac{\mu}{1+\delta} μ ′ = 1 + δ μ A : = 1 μ X M X A:=\frac{1}{\mu}XMX A := μ 1 X M X u = 1 + δ 1 n u=\sqrt{1+\delta}\mathbf{1}_n u = 1 + δ 1 n U = diag ( u ) = 1 + δ ⋅ I n U=\operatorname{diag}(u)=\sqrt{1+\delta}\cdot I_n U = diag ( u ) = 1 + δ ⋅ I n
∥ log q H x , μ ′ − log q H x , μ ∥ ∞ ≤ 16 K n log 1 + δ ≤ 8 K n δ . \lVert\log q_{H_{x,\mu'}}-\log q_{H_{x,\mu}}\rVert_\infty\le16K n\log\sqrt{1+\delta}\le8Kn\delta. ∥ log q H x , μ ′ − log q H x , μ ∥ ∞ ≤ 16 K n log 1 + δ ≤ 8 K n δ . Thus, by the range of δ \delta δ
q H x , μ ′ ≤ exp ( 8 K n δ ) ⋅ q H x , μ ≤ 2 q H x , μ . (17) q_{H_{x,\mu'}}\le\exp(8Kn\delta)\cdot q_{H_{x,\mu}}\le2q_{H_{x,\mu}}. \tag*{(17)} q H x , μ ′ ≤ exp ( 8 K n δ ) ⋅ q H x , μ ≤ 2 q H x , μ . (17) From here on to simplify notation we will denote H : = H x , μ ′ H:=H_{x,\mu'} H := H x , μ ′ H ′ : = H x ′ , μ ′ H':=H_{x',\mu'} H ′ := H x ′ , μ ′ (17) , we have that
∥ Q H err ( x , s , μ ) ∥ 2 ≤ 2 β . (18) \lVert Q_H\operatorname{err}(x,s,\mu)\rVert_2\le2\beta. \tag*{(18)} ∥ Q H err ( x , s , μ ) ∥ 2 ≤ 2 β . (18) Furthermore, one has that
err ( x , s , μ ′ ) = 1 n − x s μ ′ = 1 n − ( 1 + δ ) x s μ = ( 1 + δ ) ( 1 n − x s μ ) − δ 1 n = ( 1 + δ ) ⋅ err ( x , s , μ ) − δ 1 n . (19) \operatorname{err}(x,s,\mu')=\mathbf{1}_n-\frac{xs}{\mu'}=\mathbf{1}_n-(1+\delta)\frac{xs}{\mu}=(1+\delta)\left(\mathbf{1}_n-\frac{xs}{\mu}\right)-\delta\mathbf{1}_n=(1+\delta)\cdot\operatorname{err}(x,s,\mu)-\delta\mathbf{1}_n. \tag*{(19)} err ( x , s , μ ′ ) = 1 n − μ ′ x s = 1 n − ( 1 + δ ) μ x s = ( 1 + δ ) ( 1 n − μ x s ) − δ 1 n = ( 1 + δ ) ⋅ err ( x , s , μ ) − δ 1 n . (19) After taking a Newton step from this point, per Lemma 4.3 , we reach a new point with centrality error
err ( x ′ , s ′ , μ ′ ) = ( H err ( x , s , μ ′ ) ) 2 = ( H ( ( 1 + δ ) err ( x , s , μ ) − δ 1 n ) ) 2 . (20) \operatorname{err}(x',s',\mu')=\left(H\operatorname{err}(x,s,\mu')\right)^2=\left(H\left((1+\delta)\operatorname{err}(x,s,\mu)-\delta1_n\right)\right)^2 . \tag*{(20)} err ( x ′ , s ′ , μ ′ ) = ( H err ( x , s , μ ′ ) ) 2 = ( H ( ( 1 + δ ) err ( x , s , μ ) − δ 1 n ) ) 2 . (20) We will show that this new error has small Rump norm for our choice of δ \delta δ δ 2 ∥ Q H ′ ( H 1 n ) 2 ∥ \delta^2\lVert Q_{H'}(H1_n)^2\rVert δ 2 ∥ Q H ′ ( H 1 n ) 2 ∥ (20) and AM-GM inequality, we have
∥ Q H err ( x ′ , s ′ , μ ′ ) ∥ 2 = ∥ Q H ( ( 1 + δ ) H err ( x , s , μ ) − δ H 1 n ) 2 ∥ 2 ≤ 2 δ 2 ∥ Q H ( H 1 n ) 2 ∥ 2 + 2 ( 1 + δ ) 2 ∥ Q H ( H err ( x , s , μ ) ) 2 ∥ 2 . (21) \left\lVert Q_{H}\operatorname{err}(x',s',\mu')\right\rVert_2
=
\left\lVert Q_{H}\left((1+\delta)H\operatorname{err}(x,s,\mu)-\delta H1_n\right)^2\right\rVert_2
\le2\delta^2\left\lVert Q_{H}(H1_n)^2\right\rVert_2+2(1+\delta)^2\left\lVert Q_{H}(H\operatorname{err}(x,s,\mu))^2\right\rVert_2 . \tag*{(21)} ∥ Q H err ( x ′ , s ′ , μ ′ ) ∥ 2 = Q H ( ( 1 + δ ) H err ( x , s , μ ) − δ H 1 n ) 2 2 ≤ 2 δ 2 Q H ( H 1 n ) 2 2 + 2 ( 1 + δ ) 2 Q H ( H err ( x , s , μ ) ) 2 2 . (21) First, noting that ∥ H ∥ 2 → 2 ≤ 1 + κ \lVert H\rVert_{2\to2}\le\sqrt{1+\kappa} ∥ H ∥ 2 → 2 ≤ 1 + κ
∥ H 1 n ∥ 2 ≤ ( 1 + κ ) n (22) \lVert H1_n\rVert_2\le\sqrt{(1+\kappa)n} \tag*{(22)} ∥ H 1 n ∥ 2 ≤ ( 1 + κ ) n (22) Combining this with ∥ Q H H 1 n ∥ ∞ ≤ κ 2 2 ⋅ ( 1 + κ ) n \lVert Q_HH1_n\rVert_\infty\le\frac{\kappa^2}{2}\cdot\sqrt{(1+\kappa)n} ∥ Q H H 1 n ∥ ∞ ≤ 2 κ 2 ⋅ ( 1 + κ ) n Lemma 5.6 , we can bound the first term as
∥ Q H ( H 1 n ) 2 ∥ 2 ≤ ∥ Q H H 1 n ∥ ∞ ∥ H 1 n ∥ 2 ≤ κ 2 2 ( 1 + κ ) n . (23) \left\lVert Q_H(H1_n)^2\right\rVert_2\le\lVert Q_HH1_n\rVert_\infty\lVert H1_n\rVert_2\le\frac{\kappa^2}{2}(1+\kappa)n . \tag*{(23)} Q H ( H 1 n ) 2 2 ≤ ∥ Q H H 1 n ∥ ∞ ∥ H 1 n ∥ 2 ≤ 2 κ 2 ( 1 + κ ) n . (23) We can bound the error term by using Lemma 5.9 and (18) :
∥ Q H ( H err ( x , s , μ ) ) 2 ∥ 2 ≤ κ 2 n 3 / 2 ∥ Q H err ( x , s , μ ) ∥ 2 2 ≤ 4 κ 2 n 3 / 2 β 2 . (24) \left\lVert Q_H(H\operatorname{err}(x,s,\mu))^2\right\rVert_2\le\kappa^2n^{3/2}\lVert Q_H\operatorname{err}(x,s,\mu)\rVert_2^2\le4\kappa^2n^{3/2}\beta^2 . \tag*{(24)} Q H ( H err ( x , s , μ ) ) 2 2 ≤ κ 2 n 3/2 ∥ Q H err ( x , s , μ ) ∥ 2 2 ≤ 4 κ 2 n 3/2 β 2 . (24) Following (21) , we combine (23) and (24) to obtain
∥ Q H err ( x ′ , s ′ , μ ′ ) ∥ 2 ≤ δ 2 ⋅ κ 2 ( 1 + κ ) n + 8 ( 1 + δ ) 2 ⋅ κ 2 n 3 / 2 β 2 . \left\lVert Q_H\operatorname{err}(x',s',\mu')\right\rVert_2\le\delta^2\cdot\kappa^2(1+\kappa)n+8(1+\delta)^2\cdot\kappa^2n^{3/2}\beta^2 . ∥ Q H err ( x ′ , s ′ , μ ′ ) ∥ 2 ≤ δ 2 ⋅ κ 2 ( 1 + κ ) n + 8 ( 1 + δ ) 2 ⋅ κ 2 n 3/2 β 2 . Since 0 < δ ≤ β 1024 n ( 1 + κ ) 0<\delta\le\sqrt{\frac{\beta}{1024n(1+\kappa)}} 0 < δ ≤ 1024 n ( 1 + κ ) β 0 < β ≤ 1 16384 n 3 / 2 0<\beta\le\frac{1}{16384n^{3/2}} 0 < β ≤ 16384 n 3/2 1
∥ Q H err ( x ′ , s ′ , μ ′ ) ∥ 2 ≤ κ 2 β 1024 + 9 κ 2 β 2 n 3 / 2 ≤ β 16 + β 4 ≤ β 2 . (25) \left\lVert Q_H\operatorname{err}(x',s',\mu')\right\rVert_2\le\frac{\kappa^2\beta}{1024}+\frac{9\kappa^2\beta^2n^{3/2}}{} \le\frac{\beta}{16}+\frac{\beta}{4}\le\frac{\beta}{2}. \tag*{(25)} ∥ Q H err ( x ′ , s ′ , μ ′ ) ∥ 2 ≤ 1024 κ 2 β + 9 κ 2 β 2 n 3/2 ≤ 16 β + 4 β ≤ 2 β . (25) Finally, to bound the final Rump weights q H ′ q_{H'} q H ′ Lemma 5.7 to u = 1 n + H err ( x , s , μ ′ ) u=1_n+H\operatorname{err}(x,s,\mu') u = 1 n + H err ( x , s , μ ′ ) A = 1 μ ′ X M X A=\frac{1}{\mu'}XMX A = μ ′ 1 X M X ∣ log ( 1 + a ) ∣ ≤ 2 ∣ a ∣ |\log(1+a)|\le2|a| ∣ log ( 1 + a ) ∣ ≤ 2∣ a ∣ a ≥ − 1 / 2 a\ge-1/2 a ≥ − 1/2 ∥ log ( u ) ∥ 1 ≤ 2 ∥ H err ( x , s , μ ′ ) ∥ 1 \lVert\log(u)\rVert_1\le2\lVert H\operatorname{err}(x,s,\mu')\rVert_1 ∥ log ( u ) ∥ 1 ≤ 2 ∥ H err ( x , s , μ ′ ) ∥ 1 (19) ,
∥ H err ( x , s , μ ′ ) ∥ 1 ≤ ( 1 + δ ) ∥ H err ( x , s , μ ) ∥ 1 + δ ∥ H 1 n ∥ 1 \lVert H\operatorname{err}(x,s,\mu')\rVert_1\le(1+\delta)\lVert H\operatorname{err}(x,s,\mu)\rVert_1+\delta\lVert H1_n\rVert_1 ∥ H err ( x , s , μ ′ ) ∥ 1 ≤ ( 1 + δ ) ∥ H err ( x , s , μ ) ∥ 1 + δ ∥ H 1 n ∥ 1 We bound the first term using Lemma 5.9 and the second term from (22) , respectively. From Lemma 5.9, we get
∥ H err ( x , s , μ ) ∥ 1 ≤ ∥ Q H H err ( x , s , μ ) ∥ 1 ≤ n ∥ Q H H err ( x , s , μ ) ∥ ∞ ≤ κ n 3 / 2 ∥ Q H err ( x , s , μ ) ∥ 2 ≤ 2 κ n 3 / 2 β , \lVert H\operatorname{err}(x,s,\mu)\rVert_1\le\lVert Q_HH\operatorname{err}(x,s,\mu)\rVert_1\le n\lVert Q_HH\operatorname{err}(x,s,\mu)\rVert_\infty\le\kappa n^{3/2}\lVert Q_H\operatorname{err}(x,s,\mu)\rVert_2\le2\kappa n^{3/2}\beta, ∥ H err ( x , s , μ ) ∥ 1 ≤ ∥ Q H H err ( x , s , μ ) ∥ 1 ≤ n ∥ Q H H err ( x , s , μ ) ∥ ∞ ≤ κ n 3/2 ∥ Q H err ( x , s , μ ) ∥ 2 ≤ 2 κ n 3/2 β , and from (22) , we get ∥ H 1 n ∥ 1 ≤ n 1 + κ \lVert H1_n\rVert_1\le n\sqrt{1+\kappa} ∥ H 1 n ∥ 1 ≤ n 1 + κ δ \delta δ β \beta β
∥ H err ( x , s , μ ′ ) ∥ 1 ≤ 2 ( 1 + δ ) κ n 3 / 2 β + δ n 1 + κ ≤ 2.5 κ n 3 / 2 β + β n 1024 < 1 512 . \lVert H\operatorname{err}(x,s,\mu')\rVert_1\le2(1+\delta)\kappa n^{3/2}\beta+\delta n\sqrt{1+\kappa}\le2.5\kappa n^{3/2}\beta+\sqrt{\frac{\beta n}{1024}}<\frac{1}{512}. ∥ H err ( x , s , μ ′ ) ∥ 1 ≤ 2 ( 1 + δ ) κ n 3/2 β + δ n 1 + κ ≤ 2.5 κ n 3/2 β + 1024 β n < 512 1 . Substituting this to Lemma 5.7 , we get
∥ log q H ′ − log q H ∥ ∞ ≤ 16 κ ∥ log ( u ) ∥ 1 ≤ 32 κ ∥ H err ( x , s , μ ′ ) ∥ 1 < 1 2 , \lVert\log q_{H'}-\log q_H\rVert_\infty\le16\kappa\lVert\log(u)\rVert_1\le32\kappa\lVert H\operatorname{err}(x,s,\mu')\rVert_1<\frac{1}{2}, ∥ log q H ′ − log q H ∥ ∞ ≤ 16 κ ∥ log ( u ) ∥ 1 ≤ 32 κ ∥ H err ( x , s , μ ′ ) ∥ 1 < 2 1 , so q H ′ ≤ exp ( 1 / 2 ) ⋅ q H ≤ 2 q H q_{H'}\le\exp(1/2)\cdot q_H\le2q_H q H ′ ≤ exp ( 1/2 ) ⋅ q H ≤ 2 q H (25) , we get ∥ Q H ′ err ( x ′ , s ′ , μ ′ ) ∥ 2 ≤ 2 ∥ Q H err ( x ′ , s ′ , μ ′ ) ∥ 2 ≤ β \lVert Q_{H'}\operatorname{err}(x',s',\mu')\rVert_2\le2\lVert Q_H\operatorname{err}(x',s',\mu')\rVert_2\le\beta ∥ Q H ′ err ( x ′ , s ′ , μ ′ ) ∥ 2 ≤ 2 ∥ Q H err ( x ′ , s ′ , μ ′ ) ∥ 2 ≤ β
We are now ready to prove Theorem 5.1.
Proof of Theorem 5.1. We show inductively that for the specified choices of β \beta β δ \delta δ N R ( β ) \mathcal{N}^{R}(\beta) N R ( β ) ( x 0 , s 0 , μ 0 ) (x^0,s^0,\mu_0) ( x 0 , s 0 , μ 0 ) ( x , s , μ ) ∈ N R ( β ) (x,s,\mu)\in\mathcal{N}^{R}(\beta) ( x , s , μ ) ∈ N R ( β ) μ ′ = μ / ( 1 + δ ) \mu'=\mu/(1+\delta) μ ′ = μ / ( 1 + δ ) μ ′ \mu' μ ′ β \beta β δ \delta δ Lemma 5.10 , we get ( x ′ , s ′ , μ ′ ) ∈ N R ( β ) (x',s',\mu')\in\mathcal{N}^{R}(\beta) ( x ′ , s ′ , μ ′ ) ∈ N R ( β ) Lemma 5.10 , completing the induction. By the choice of δ \delta δ O ( δ − 1 log ( μ 0 / μ 1 ) ) = O ( n 5 / 4 1 + κ log ( μ 0 / μ 1 ) ) O(\delta^{-1}\log(\mu_0/\mu_1))=O(n^{5/4}\sqrt{1+\kappa}\log(\mu_0/\mu_1)) O ( δ − 1 log ( μ 0 / μ 1 )) = O ( n 5/4 1 + κ log ( μ 0 / μ 1 )) q H q_H q H
Bounding the Rump weights
In what follows, we derive the technical lemmas needed to prove Lemma 5.6 . We start with a simple corollary of Theorem 1.1.
Corollary 5.11. Let A ∈ P ⋆ ( κ ) A \in\mathcal{P}_{\star}(\kappa) A ∈ P ⋆ ( κ ) D ∈ D > 0 n D \in\mathcal{D}_{>0}^{n} D ∈ D > 0 n
∥ ( D + A ) − 1 ∥ 2 ≤ 1 + κ min i ∈ [ n ] d i . \left\|(D+A)^{-1}\right\|_{2} \leq\frac{\sqrt{1+\kappa}}{\min_{i\in[n]}d_i}. ( D + A ) − 1 2 ≤ min i ∈ [ n ] d i 1 + κ . Proof. We write D + A = D 1 / 2 ( I n + D − 1 / 2 A D − 1 / 2 ) D 1 / 2 D+A=D^{1/2}(I_n+D^{-1/2}AD^{-1/2})D^{1/2} D + A = D 1/2 ( I n + D − 1/2 A D − 1/2 ) D 1/2 1 + κ \sqrt{1+\kappa} 1 + κ
∥ ( D + A ) − 1 ∥ 2 ≤ ∥ D − 1 / 2 ∥ 2 2 1 + κ ≤ 1 + κ min i ∈ [ n ] d i . \left\|(D+A)^{-1}\right\|_{2}\leq\left\|D^{-1/2}\right\|_{2}^{2}\sqrt{1+\kappa}\leq\frac{\sqrt{1+\kappa}}{\min_{i\in[n]}d_i}. ( D + A ) − 1 2 ≤ D − 1/2 2 2 1 + κ ≤ min i ∈ [ n ] d i 1 + κ . We use this to derive the next technical lemma that shows stability of ρ 0 S ( ( I + A ) − 1 ) \rho^S_0((I+A)^{-1}) ρ 0 S (( I + A ) − 1 )
Lemma 5.12. Let A ∈ P ⋆ ( κ ) A\in\mathcal{P}_{\star}(\kappa) A ∈ P ⋆ ( κ ) H : = ( I n + A ) − 1 H:=(I_n+A)^{-1} H := ( I n + A ) − 1 u ∈ R n u\in\mathbb{R}^n u ∈ R n k ∈ [ n ] k\in[n] k ∈ [ n ] M ∈ R n × n M\in\mathbb{R}^{n\times n} M ∈ R n × n k k k H H H H u Hu H u M ∙ k = H u M_{\bullet k}=Hu M ∙ k = H u M ∙ i = H ∙ i M_{\bullet i}=H_{\bullet i} M ∙ i = H ∙ i i ≠ k i\ne k i = k
ρ 0 S ( M ) ≤ 1 + 1 + κ ⋅ ∥ u ∥ 2 . \rho^S_0(M)\leq1+\sqrt{1+\kappa}\cdot\|u\|_2. ρ 0 S ( M ) ≤ 1 + 1 + κ ⋅ ∥ u ∥ 2 . Proof. The claim is trivial if ρ 0 S ( M ) ≤ 1 \rho^S_0(M)\leq1 ρ 0 S ( M ) ≤ 1 ρ 0 S ( M ) > 1 \rho^S_0(M)>1 ρ 0 S ( M ) > 1 λ : = ρ 0 S ( M ) \lambda:=\rho^S_0(M) λ := ρ 0 S ( M ) S ∈ diag { ± 1 } S\in\operatorname{diag}\{\pm1\} S ∈ diag { ± 1 } v ∈ R n v\in\mathbb{R}^n v ∈ R n S M v = λ v SMv=\lambda v S M v = λ v v ′ = S v v'=Sv v ′ = S v M S v ′ = λ v ′ MSv'=\lambda v' M S v ′ = λ v ′ H − 1 = ( I + A ) H^{-1}=(I+A) H − 1 = ( I + A )
( I n + A − 1 λ H − 1 M S ) v ′ = 0 n . \left(I_n+A-\frac{1}{\lambda}H^{-1}MS\right)v'=0_n. ( I n + A − λ 1 H − 1 M S ) v ′ = 0 n . Note that ( H − 1 M ) ∙ i = e i (H^{-1}M)_{\bullet i}=e_i ( H − 1 M ) ∙ i = e i i ≠ k i\ne k i = k ( H − 1 M ) ∙ k = H − 1 H u = u (H^{-1}M)_{\bullet k}=H^{-1}Hu=u ( H − 1 M ) ∙ k = H − 1 H u = u D ∈ R n × n D\in\mathbb{R}^{n\times n} D ∈ R n × n D i i : = 1 − S i i λ > 0 D_{ii}:=1-\frac{S_{ii}}{\lambda}>0 D ii := 1 − λ S ii > 0 i ≠ k i\ne k i = k D k k : = 1 D_{kk}:=1 D k k := 1
( D + A ) v ′ = S k k v k ′ λ ⋅ u , (D+A)v'=\frac{S_{kk}v'_k}{\lambda}\cdot u, ( D + A ) v ′ = λ S k k v k ′ ⋅ u , that is,
v ′ = S k k v k ′ λ ⋅ ( D + A ) − 1 u . v'=\frac{S_{kk}v'_k}{\lambda}\cdot(D+A)^{-1}u. v ′ = λ S k k v k ′ ⋅ ( D + A ) − 1 u . By Corollary 5.11, we have ∥ ( D + A ) − 1 ∥ 2 ≤ λ λ − 1 ⋅ 1 + κ \left\|(D+A)^{-1}\right\|_2\leq\frac{\lambda}{\lambda-1}\cdot\sqrt{1+\kappa} ( D + A ) − 1 2 ≤ λ − 1 λ ⋅ 1 + κ
∥ v ′ ∥ 2 ≤ ∣ S k k v k ′ ∣ λ ⋅ ∥ ( D + A ) − 1 ∥ 2 ⋅ ∥ u ∥ 2 ≤ ∣ v k ′ ∣ λ − 1 ⋅ 1 + κ ⋅ ∥ u ∥ 2 , \|v'\|_2\leq\frac{|S_{kk}v'_k|}{\lambda}\cdot\left\|(D+A)^{-1}\right\|_2\cdot\|u\|_2\leq\frac{|v'_k|}{\lambda-1}\cdot\sqrt{1+\kappa}\cdot\|u\|_2, ∥ v ′ ∥ 2 ≤ λ ∣ S k k v k ′ ∣ ⋅ ( D + A ) − 1 2 ⋅ ∥ u ∥ 2 ≤ λ − 1 ∣ v k ′ ∣ ⋅ 1 + κ ⋅ ∥ u ∥ 2 , implying 1 ≤ 1 λ − 1 ⋅ 1 + κ ⋅ ∥ u ∥ 2 1\leq\frac{1}{\lambda-1}\cdot\sqrt{1+\kappa}\cdot\|u\|_2 1 ≤ λ − 1 1 ⋅ 1 + κ ⋅ ∥ u ∥ 2 λ ≤ 1 + 1 + κ ∥ u ∥ 2 \lambda\leq1+\sqrt{1+\kappa}\|u\|_2 λ ≤ 1 + 1 + κ ∥ u ∥ 2
We can now prove Lemma 5.6.
Lemma 5.6 (Bounding Rump weights). For A ∈ P ⋆ ( κ ) A\in\mathcal{P}_{\star}(\kappa) A ∈ P ⋆ ( κ ) H = ( I n + A ) − 1 H=(I_n+A)^{-1} H = ( I n + A ) − 1 Q H = diag ( q H ) Q_H=\operatorname{diag}(q_H) Q H = diag ( q H ) Definition 5.2 . Then, ∥ Q H H v ∥ ∞ ≤ K 2 2 1 + κ ∥ v ∥ 2 \|Q_HHv\|_{\infty}\leq\frac{K^2}{2}\sqrt{1+\kappa}\|v\|_2 ∥ Q H H v ∥ ∞ ≤ 2 K 2 1 + κ ∥ v ∥ 2 v ∈ R n v\in\mathbb{R}^n v ∈ R n 1 n ≤ q H ≤ K 2 1 + κ ⋅ 1 n \mathbf{1}_n\leq q_H\leq\frac{K}{2}\sqrt{1+\kappa}\cdot\mathbf{1}_n 1 n ≤ q H ≤ 2 K 1 + κ ⋅ 1 n
Proof. We can w.l.o.g. assume that ∥ v ∥ 2 = 1 \|v\|_2=1 ∥ v ∥ 2 = 1 τ : = 1 + κ \tau:=\sqrt{1+\kappa} τ := 1 + κ
∥ Q H H v ∥ ∞ ≤ K 2 2 ⋅ τ ⋅ ∥ v ∥ 2 , ∀ v ∈ R n . (26) \|Q_HHv\|_{\infty}\leq\frac{K^2}{2}\cdot\tau\cdot\|v\|_2,\qquad\forall v\in\mathbb{R}^n. \tag*{(26)} ∥ Q H H v ∥ ∞ ≤ 2 K 2 ⋅ τ ⋅ ∥ v ∥ 2 , ∀ v ∈ R n . (26) Let us fix some index ℓ ∈ [ n ] \ell\in[n] ℓ ∈ [ n ] q H ( ℓ ) ∣ ( H v ) ℓ ∣ ≤ K 2 τ / 2 q_H(\ell)|(Hv)_{\ell}|\leq K^2\tau/2 q H ( ℓ ) ∣ ( H v ) ℓ ∣ ≤ K 2 τ /2 ( q H ) ℓ ≥ 1 (q_H)_{\ell}\geq1 ( q H ) ℓ ≥ 1 q H ( ℓ ) = 1 q_H(\ell)=1 q H ( ℓ ) = 1 ∥ H v ∥ 2 ≤ τ \|Hv\|_2\leq\tau ∥ H v ∥ 2 ≤ τ q H ( ℓ ) q_H(\ell) q H ( ℓ ) w i j = ∣ H i j ∣ / K w_{ij}=|H_{ij}|/K w ij = ∣ H ij ∣/ K P P P ℓ \ell ℓ k k k M M M k k k H H H 2 H v / ( K τ ) 2Hv/(K\tau) 2 H v / ( K τ ) Lemma 5.12 , with u = 2 v / ( K τ ) u=2v/(K\tau) u = 2 v / ( K τ ) ∥ v ∥ 2 = 1 \lVert v\rVert_2=1 ∥ v ∥ 2 = 1
ρ 0 S ( M ) ≤ 1 + τ ∥ u ∥ 2 ≤ 1 + 2 K . \rho_0^S(M) \le1+\tau\lVert u\rVert_2 \le1+\frac{2}{K}. ρ 0 S ( M ) ≤ 1 + τ ∥ u ∥ 2 ≤ 1 + K 2 . Then using Theorem 2.10 , ζ ( M ) ≤ ( K − 2 ) ρ 0 S ( M ) ≤ ( K − 2 ) ( 1 + 2 / K ) < K \zeta(M)\le(K-2)\rho_0^S(M)\le(K-2)(1+2/K)<K ζ ( M ) ≤ ( K − 2 ) ρ 0 S ( M ) ≤ ( K − 2 ) ( 1 + 2/ K ) < K w i j ′ = ∣ M i j ∣ / K w'_{ij}=|M_{ij}|/K w ij ′ = ∣ M ij ∣/ K i ∈ [ n ] i\in[n] i ∈ [ n ] w i j ′ = w i j w'_{ij}=w_{ij} w ij ′ = w ij j ≠ k j\ne k j = k w i k ′ : = 2 ∣ ( H v ) i ∣ / ( K 2 τ ) w'_{ik}:=2|(Hv)_i|/(K^2\tau) w ik ′ := 2∣ ( H v ) i ∣/ ( K 2 τ ) C C C k − ℓ k-\ell k − ℓ P P P ( ℓ , k ) (\ell,k) ( ℓ , k )
1 > ζ ( M ) K ≥ ∏ e ∈ C w e ′ = 2 ∣ ( H v ) ℓ ∣ K 2 τ ⋅ ∏ e ∈ P w e = 2 ∣ ( H v ) ℓ ∣ K 2 τ ⋅ q H ( ℓ ) . 1>\frac{\zeta(M)}{K}\ge\prod_{e\in C}w'_e=\frac{2|(Hv)_\ell|}{K^2\tau}\cdot\prod_{e\in P}w_e=\frac{2|(Hv)_\ell|}{K^2\tau}\cdot q_H(\ell). 1 > K ζ ( M ) ≥ e ∈ C ∏ w e ′ = K 2 τ 2∣ ( H v ) ℓ ∣ ⋅ e ∈ P ∏ w e = K 2 τ 2∣ ( H v ) ℓ ∣ ⋅ q H ( ℓ ) . Thus, q H ( ℓ ) ∣ ( H v ) ℓ ∣ ≤ K 2 τ / 2 q_H(\ell)|(Hv)_\ell|\le K^2\tau/2 q H ( ℓ ) ∣ ( H v ) ℓ ∣ ≤ K 2 τ /2
The first part of the lemma follows since ∥ Q H H v ∥ ∞ = max ℓ ∈ [ n ] q H ( ℓ ) ∣ ( H v ) ℓ ∣ ≤ K 2 2 τ \lVert Q_HHv\rVert_\infty=\max_{\ell\in[n]}q_H(\ell)|(Hv)_\ell|\le\frac{K^2}{2}\tau ∥ Q H H v ∥ ∞ = max ℓ ∈ [ n ] q H ( ℓ ) ∣ ( H v ) ℓ ∣ ≤ 2 K 2 τ ℓ ∈ [ n ] \ell\in[n] ℓ ∈ [ n ] q H ( ℓ ) ≥ 1 q_H(\ell)\ge1 q H ( ℓ ) ≥ 1 q H ( ℓ ) > 1 q_H(\ell)>1 q H ( ℓ ) > 1 q H ( ℓ ) = q H ( j ) ∣ H j ℓ ∣ / K q_H(\ell)=q_H(j)|H_{j\ell}|/K q H ( ℓ ) = q H ( j ) ∣ H j ℓ ∣/ K j ≠ ℓ j\ne\ell j = ℓ v = e ℓ v=e_\ell v = e ℓ (26) yields
K 2 2 τ ≥ q H ( j ) ∣ ( H v ) j ∣ = q H ( j ) ∣ H j ℓ ∣ = K q H ( ℓ ) , \frac{K^2}{2}\tau\ge q_H(j)|(Hv)_j|=q_H(j)|H_{j\ell}|=Kq_H(\ell), 2 K 2 τ ≥ q H ( j ) ∣ ( H v ) j ∣ = q H ( j ) ∣ H j ℓ ∣ = K q H ( ℓ ) , which implies q H ( ℓ ) ≤ K τ / 2 q_H(\ell)\le K\tau/2 q H ( ℓ ) ≤ K τ /2 ℓ ∈ [ n ] \ell\in[n] ℓ ∈ [ n ] q H ≤ K 2 τ 1 n q_H\le\frac{K}{2}\tau\mathbf{1}_n q H ≤ 2 K τ 1 n
Stability of Rump weights
Lemma 5.7. (Stability of Rump weights ). For A ∈ P ⋆ ∩ R n × n A\in\mathcal{P}_\star\cap\mathbb{R}^{n\times n} A ∈ P ⋆ ∩ R n × n u ∈ R > 0 n u\in\mathbb{R}^n_{>0} u ∈ R > 0 n U = diag ( u ) U=\operatorname{diag}(u) U = diag ( u ) H : = ( I n + A ) − 1 H:=(I_n+A)^{-1} H := ( I n + A ) − 1 H ′ : = ( I n + U A U ) − 1 H':=(I_n+UAU)^{-1} H ′ := ( I n + U A U ) − 1 q H , q H ′ q_H,q_{H'} q H , q H ′ Definition 5.2 , we have
∥ log q H ′ − log q H ∥ ∞ ≤ 16 K ∥ log ( u ) ∥ 1 . (27) \lVert\log q_{H'}-\log q_H\rVert_\infty\le16K\lVert\log(u)\rVert_1. \tag*{(27)} ∥ log q H ′ − log q H ∥ ∞ ≤ 16 K ∥ log ( u ) ∥ 1 . (27) To prove the lemma, we will rely on a few helper statements. Let z : = log u z:=\log u z := log u Z : = diag ( z ) Z:=\operatorname{diag}(z) Z := diag ( z ) t ∈ [ 0 , 1 ] t\in[0,1] t ∈ [ 0 , 1 ] A ( t ) : = e t Z A e t Z A(t):=e^{tZ}Ae^{tZ} A ( t ) := e tZ A e tZ H ( t ) : = ( I n + A ( t ) ) − 1 H(t):=(I_n+A(t))^{-1} H ( t ) := ( I n + A ( t ) ) − 1 d ( t ) : = q H ( t ) d(t):=q_{H(t)} d ( t ) := q H ( t )
Since A ( t ) A(t) A ( t ) H ( t ) H(t) H ( t ) H ( 0 ) = H H(0)=H H ( 0 ) = H H ( 1 ) = H ′ H(1)=H' H ( 1 ) = H ′ H ( 1 ) H(1) H ( 1 ) H ( t ) H(t) H ( t ) A ( t ) A(t) A ( t )
A ˙ ( t ) = Z A ( t ) + A ( t ) Z . (28) \dot{A}(t)=ZA(t)+A(t)Z. \tag*{(28)} A ˙ ( t ) = Z A ( t ) + A ( t ) Z . (28) Using this fact we can study the change of H ( t ) H(t) H ( t )
Claim 5.13. The matrix H ( t ) H(t) H ( t ) t ∈ [ 0 , 1 ] t\in[0,1] t ∈ [ 0 , 1 ]
H ˙ ( t ) = − H ( t ) Z − Z H ( t ) + 2 H ( t ) Z H ( t ) , for all t ∈ [ 0 , 1 ] . \dot{H}(t)=-H(t)Z-ZH(t)+2H(t)ZH(t),\qquad\text{for all }t\in[0,1]. H ˙ ( t ) = − H ( t ) Z − Z H ( t ) + 2 H ( t ) Z H ( t ) , for all t ∈ [ 0 , 1 ] . Proof. Recalling that H ( t ) = ( I n + A ( t ) ) − 1 H(t)=(I_n+A(t))^{-1} H ( t ) = ( I n + A ( t ) ) − 1 ( I n + A ( t ) ) H ( t ) = I n (I_n+A(t))H(t)=I_n ( I n + A ( t )) H ( t ) = I n A ˙ ( t ) H ( t ) + ( I n + A ( t ) ) H ˙ ( t ) = 0 \dot{A}(t)H(t)+(I_n+A(t))\dot{H}(t)=0 A ˙ ( t ) H ( t ) + ( I n + A ( t )) H ˙ ( t ) = 0 (28) , we have
H ˙ ( t ) = − ( I n + A ( t ) ) − 1 A ˙ ( t ) H ( t ) = − H ( t ) A ˙ ( t ) H ( t ) = − H ( t ) ( Z A ( t ) + A ( t ) Z ) H ( t ) . \dot{H}(t)=-(I_n+A(t))^{-1}\dot{A}(t)H(t)=-H(t)\dot{A}(t)H(t)=-H(t)(ZA(t)+A(t)Z)H(t). H ˙ ( t ) = − ( I n + A ( t ) ) − 1 A ˙ ( t ) H ( t ) = − H ( t ) A ˙ ( t ) H ( t ) = − H ( t ) ( Z A ( t ) + A ( t ) Z ) H ( t ) . Since A ( t ) H ( t ) = H ( t ) A ( t ) = I n − H ( t ) A(t)H(t)=H(t)A(t)=I_n-H(t) A ( t ) H ( t ) = H ( t ) A ( t ) = I n − H ( t )
H ˙ ( t ) = − H ( t ) Z A ( t ) H ( t ) − H ( t ) A ( t ) Z H ( t ) = − H ( t ) Z ( I n − H ( t ) ) − ( I n − H ( t ) ) Z H ( t ) = − H ( t ) Z − Z H ( t ) + 2 H ( t ) Z H ( t ) , (29) \begin{aligned}
\dot{H}(t)&=-H(t)ZA(t)H(t)-H(t)A(t)ZH(t)\\
&=-H(t)Z(I_n-H(t))-(I_n-H(t))ZH(t)\\
&=-H(t)Z-ZH(t)+2H(t)ZH(t), \tag*{(29)}
\end{aligned} H ˙ ( t ) = − H ( t ) Z A ( t ) H ( t ) − H ( t ) A ( t ) Z H ( t ) = − H ( t ) Z ( I n − H ( t )) − ( I n − H ( t )) Z H ( t ) = − H ( t ) Z − Z H ( t ) + 2 H ( t ) Z H ( t ) , (29) as needed.
We write w P ( t ) w_P(t) w P ( t ) P P P Definition 5.2 :
w P ( t ) : = ∏ r = 1 ℓ ∣ H i r i r + 1 ( t ) ∣ K , P = ( i 1 , … , i ℓ + 1 = j ) . w_P(t):=\prod_{r=1}^{\ell}\frac{|H_{i_ri_{r+1}}(t)|}{K},\qquad P=(i_1,\ldots,i_{\ell+1}=j). w P ( t ) := r = 1 ∏ ℓ K ∣ H i r i r + 1 ( t ) ∣ , P = ( i 1 , … , i ℓ + 1 = j ) . By that definition, d j ( t ) = max P w P ( t ) d_j(t)=\max_P w_P(t) d j ( t ) = max P w P ( t ) j j j 1 1 1
Proposition 5.14. For every j ∈ [ n ] j \in[n] j ∈ [ n ] d j d_j d j log d j \log d_j log d j [ 0 , 1 ] [0,1] [ 0 , 1 ] t ∈ ( 0 , 1 ) t \in(0,1) t ∈ ( 0 , 1 ) d j d_j d j P P P
d ˙ j ( t ) = w ˙ P ( t ) . \dot{d}_j(t)=\dot{w}_P(t). d ˙ j ( t ) = w ˙ P ( t ) . In particular, d ˙ j ( t ) = 0 \dot{d}_j(t)=0 d ˙ j ( t ) = 0 d j ( t ) = 1 d_j(t)=1 d j ( t ) = 1
Proof. Each matrix entry is continuously differentiable on [ 0 , 1 ] [0,1] [ 0 , 1 ] w P w_P w P d j d_j d j log d j \log d_j log d j d j ≥ 1 d_j \ge1 d j ≥ 1 t ∈ ( 0 , 1 ) t \in(0,1) t ∈ ( 0 , 1 ) P P P w P ( t ) = d j ( t ) ≥ 1 w_P(t)=d_j(t)\ge1 w P ( t ) = d j ( t ) ≥ 1 t t t w P w_P w P d j − w P d_j-w_P d j − w P t t t d j ( t ) = 1 d_j(t)=1 d j ( t ) = 1
The following bound controls the derivative along a maximizing path.
Lemma 5.15. Let t ∈ [ 0 , 1 ] t \in[0,1] t ∈ [ 0 , 1 ] P = ( i 1 , … , i ℓ + 1 = j ) P=(i_1,\ldots,i_{\ell+1}=j) P = ( i 1 , … , i ℓ + 1 = j ) d j ( t ) = w P ( t ) d_j(t)=w_P(t) d j ( t ) = w P ( t ) r ∈ [ ℓ ] r \in[\ell] r ∈ [ ℓ ]
d i r ( t ) d i r + 1 ( t ) ∣ H i r i r + 1 ( t ) ∣ = K . \frac{d_{i_r}(t)}{d_{i_{r+1}}(t)}\left|H_{i_ri_{r+1}}(t)\right|=K. d i r + 1 ( t ) d i r ( t ) ∣ H i r i r + 1 ( t ) ∣ = K . Furthermore, for every k ∈ [ n ] k \in[n] k ∈ [ n ]
∑ r = 1 ℓ d i r ( t ) d i r + 1 ( t ) ∣ H i r k ( t ) H k i r + 1 ( t ) ∣ ≤ 7 K 2 . \sum_{r=1}^{\ell}\frac{d_{i_r}(t)}{d_{i_{r+1}}(t)}\left|H_{i_rk}(t)H_{ki_{r+1}}(t)\right|\le7K^2. r = 1 ∑ ℓ d i r + 1 ( t ) d i r ( t ) ∣ H i r k ( t ) H k i r + 1 ( t ) ∣ ≤ 7 K 2 . We first prove the combinatorial bound needed for this lemma.
Lemma 5.16. Let G = ( V , E , ρ ) G=(V,E,\rho) G = ( V , E , ρ ) ρ ∈ [ 0 , 1 ] E \rho\in[0,1]^E ρ ∈ [ 0 , 1 ] E C C C
∏ e ∈ C ρ ( e ) ≤ ( 3 / 4 ) ∣ C ∣ . \prod_{e\in C}\rho(e)\le(3/4)^{|C|}. e ∈ C ∏ ρ ( e ) ≤ ( 3/4 ) ∣ C ∣ . Let P = ( i 1 , … , i ℓ ) P=(i_1,\ldots,i_\ell) P = ( i 1 , … , i ℓ ) 1 1 1 r ∈ V r \in V r ∈ V
∑ j = 1 ℓ − 1 ρ ( i j , r ) ρ ( r , i j + 1 ) ≤ 7. \sum_{j=1}^{\ell-1}\rho(i_j,r)\rho(r,i_{j+1})\le7. j = 1 ∑ ℓ − 1 ρ ( i j , r ) ρ ( r , i j + 1 ) ≤ 7. Proof. Let a j : = ρ ( i j , r ) a_j:=\rho(i_j,r) a j := ρ ( i j , r ) b j : = ρ ( r , i j ) b_j:=\rho(r,i_j) b j := ρ ( r , i j ) ( r , i k , … , i j , r ) (r,i_k,\ldots,i_j,r) ( r , i k , … , i j , r )
a j b k ≤ ( 3 / 4 ) j − k + 2 ( 1 ≤ k ≤ j ≤ ℓ ) . (30) a_jb_k\le(3/4)^{j-k+2}\qquad(1\le k\le j\le\ell). \tag*{(30)} a j b k ≤ ( 3/4 ) j − k + 2 ( 1 ≤ k ≤ j ≤ ℓ ) . (30) This remains valid when r r r P P P
Let t ∈ { 1 , … , ℓ } t\in\{1,\ldots,\ell\} t ∈ { 1 , … , ℓ } a j ≤ ( 3 / 4 ) j − t + 1 a_j\le(3/4)^{j-t+1} a j ≤ ( 3/4 ) j − t + 1 t ≤ j ≤ ℓ − 1 t\le j\le\ell-1 t ≤ j ≤ ℓ − 1 t t t t = ℓ t=\ell t = ℓ
∑ j = t ℓ − 1 a j b j + 1 ≤ ∑ h = 1 ∞ ( 3 / 4 ) h = 3. \sum_{j=t}^{\ell-1}a_jb_{j+1}\le\sum_{h=1}^{\infty}(3/4)^h=3. j = t ∑ ℓ − 1 a j b j + 1 ≤ h = 1 ∑ ∞ ( 3/4 ) h = 3. If t = 1 t=1 t = 1 j ⋆ ∈ { t − 1 , … , ℓ − 1 } j^\star\in\{t-1,\ldots,\ell-1\} j ⋆ ∈ { t − 1 , … , ℓ − 1 } a j ⋆ > ( 3 / 4 ) j ⋆ − t + 2 a_{j^\star}>(3/4)^{j^\star-t+2} a j ⋆ > ( 3/4 ) j ⋆ − t + 2 (30) , b k ≤ ( 3 / 4 ) t − k b_k\le(3/4)^{t-k} b k ≤ ( 3/4 ) t − k k ≤ t − 1 k\le t-1 k ≤ t − 1
∑ j = 1 t − 2 a j b j + 1 ≤ ∑ h = 1 ∞ ( 3 / 4 ) h = 3. \sum_{j=1}^{t-2}a_jb_{j+1}\le\sum_{h=1}^{\infty}(3/4)^h=3. j = 1 ∑ t − 2 a j b j + 1 ≤ h = 1 ∑ ∞ ( 3/4 ) h = 3. These two bounds cover all terms except j = t − 1 j=t-1 j = t − 1 a t − 1 , b t ∈ [ 0 , 1 ] a_{t-1},b_t\in[0,1] a t − 1 , b t ∈ [ 0 , 1 ]
∑ j = 1 ℓ − 1 a j b j + 1 = ∑ j = 1 t − 2 a j b j + 1 + a t − 1 b t + ∑ j = t ℓ − 1 a j b j + 1 ≤ 3 + 1 + 3 = 7. \sum_{j=1}^{\ell-1}a_jb_{j+1}
=\sum_{j=1}^{t-2}a_jb_{j+1}+a_{t-1}b_t+\sum_{j=t}^{\ell-1}a_jb_{j+1}
\le3+1+3=7. j = 1 ∑ ℓ − 1 a j b j + 1 = j = 1 ∑ t − 2 a j b j + 1 + a t − 1 b t + j = t ∑ ℓ − 1 a j b j + 1 ≤ 3 + 1 + 3 = 7. Now we are ready to prove Lemma 5.15 , which we will then use to conclude the main stability lemma.
Proof of Lemma 5.15. To simplify notation, for the rest of the proof we will drop the time parameter t t t ρ ( a , b ) : = ( d a / d b ) ⋅ ∣ H a b ∣ / K \rho(a,b):=(d_a/d_b)\cdot|H_{ab}|/K ρ ( a , b ) := ( d a / d b ) ⋅ ∣ H ab ∣/ K a , b ∈ [ n ] a,b\in[n] a , b ∈ [ n ] Lemma 5.4 , these weights lie in [ 0 , 1 ] [0,1] [ 0 , 1 ] P P P d j d_j d j
∏ r = 1 ℓ ρ ( i r , i r + 1 ) = d i 1 d j w P = d i 1 ≥ 1. \prod_{r=1}^{\ell}\rho(i_r,i_{r+1})=\frac{d_{i_1}}{d_j}w_P=d_{i_1}\ge1. r = 1 ∏ ℓ ρ ( i r , i r + 1 ) = d j d i 1 w P = d i 1 ≥ 1. Every factor is at most 1 1 1 P P P 1 1 1 C C C d a / d b d_a/d_b d a / d b Lemma 5.4 , with K = 8 K=8 K = 8
∏ ( a , b ) ∈ C ρ ( a , b ) = ∏ ( a , b ) ∈ C ∣ H a b ∣ K ≤ ( ζ ( H ) K ) ∣ C ∣ ≤ ( 3 / 4 ) ∣ C ∣ . \prod_{(a,b)\in C}\rho(a,b)=\prod_{(a,b)\in C}\frac{|H_{ab}|}{K}\le\left(\frac{\zeta(H)}{K}\right)^{|C|}\le(3/4)^{|C|}. ( a , b ) ∈ C ∏ ρ ( a , b ) = ( a , b ) ∈ C ∏ K ∣ H ab ∣ ≤ ( K ζ ( H ) ) ∣ C ∣ ≤ ( 3/4 ) ∣ C ∣ . Applying Lemma 5.16 with r = k r=k r = k
∑ r = 1 ℓ d i r d i r + 1 ∣ H i r k H k i r + 1 ∣ = K 2 ∑ r = 1 ℓ ρ ( i r , k ) ρ ( k , i r + 1 ) ≤ 7 K 2 . \sum_{r=1}^{\ell}\frac{d_{i_r}}{d_{i_{r+1}}}|H_{i_rk}H_{ki_{r+1}}|=K^2\sum_{r=1}^{\ell}\rho(i_r,k)\rho(k,i_{r+1})\le7K^2. r = 1 ∑ ℓ d i r + 1 d i r ∣ H i r k H k i r + 1 ∣ = K 2 r = 1 ∑ ℓ ρ ( i r , k ) ρ ( k , i r + 1 ) ≤ 7 K 2 . We are now ready to prove the main statement of this section, Lemma 5.7 .
Proof of Lemma 5.7. Let j ∈ [ n ] j\in[n] j ∈ [ n ] t ∈ ( 0 , 1 ) t\in(0,1) t ∈ ( 0 , 1 ) d j d_j d j t t t d j ( t ) = 1 d_j(t)=1 d j ( t ) = 1 ( log d j ) ′ ( t ) = 0 (\log d_j)'(t)=0 ( log d j ) ′ ( t ) = 0 Proposition 5.14 . Otherwise, there is a non-empty maximizing path P = ( i 1 , … , i ℓ + 1 = j ) P=(i_1,\ldots,i_{\ell+1}=j) P = ( i 1 , … , i ℓ + 1 = j ) H H H P P P Claim 5.13 , H ˙ ( t ) = − H ( t ) Z − Z H ( t ) + 2 H ( t ) Z H ( t ) \dot{H}(t)=-H(t)Z-ZH(t)+2H(t)ZH(t) H ˙ ( t ) = − H ( t ) Z − Z H ( t ) + 2 H ( t ) Z H ( t ) Z Z Z t t t r ∈ [ ℓ ] r\in[\ell] r ∈ [ ℓ ] ( Z H ) i r i r + 1 = z i r H i r i r + 1 (ZH)_{i_ri_{r+1}}=z_{i_r}H_{i_ri_{r+1}} ( Z H ) i r i r + 1 = z i r H i r i r + 1 ( H Z ) i r i r + 1 = z i r + 1 H i r i r + 1 (HZ)_{i_ri_{r+1}}=z_{i_{r+1}}H_{i_ri_{r+1}} ( H Z ) i r i r + 1 = z i r + 1 H i r i r + 1 ( H Z H ) i r i r + 1 = ∑ k = 1 n z k H i r k H k i r + 1 (HZH)_{i_ri_{r+1}}=\sum_{k=1}^{n}z_kH_{i_rk}H_{ki_{r+1}} ( H Z H ) i r i r + 1 = ∑ k = 1 n z k H i r k H k i r + 1 H i r i r + 1 ≠ 0 H_{i_ri_{r+1}}\ne0 H i r i r + 1 = 0 r ∈ [ ℓ ] r\in[\ell] r ∈ [ ℓ ]
H ˙ i r i r + 1 H i r i r + 1 = − z i r − z i r + 1 + 2 ∑ k = 1 n z k ⋅ H i r k H k i r + 1 H i r i r + 1 . \frac{\dot{H}_{i_ri_{r+1}}}{H_{i_ri_{r+1}}}=-z_{i_r}-z_{i_{r+1}}+2\sum_{k=1}^{n}z_k\cdot\frac{H_{i_rk}H_{ki_{r+1}}}{H_{i_ri_{r+1}}}. H i r i r + 1 H ˙ i r i r + 1 = − z i r − z i r + 1 + 2 k = 1 ∑ n z k ⋅ H i r i r + 1 H i r k H k i r + 1 . Using ( log ∣ H i r i r + 1 ∣ ) ′ = H ˙ i r i r + 1 H i r i r + 1 (\log|H_{i_ri_{r+1}}|)'=\frac{\dot{H}_{i_ri_{r+1}}}{H_{i_ri_{r+1}}} ( log ∣ H i r i r + 1 ∣ ) ′ = H i r i r + 1 H ˙ i r i r + 1 H i r i r + 1 ≠ 0 H_{i_ri_{r+1}}\ne0 H i r i r + 1 = 0 log w P = ∑ r = 1 ℓ log ∣ H i r i r + 1 ∣ − ℓ log K \log w_P=\sum_{r=1}^{\ell}\log|H_{i_ri_{r+1}}|-\ell\log K log w P = ∑ r = 1 ℓ log ∣ H i r i r + 1 ∣ − ℓ log K
( log w P ) ′ = ∑ r = 1 ℓ H ˙ i r i r + 1 H i r i r + 1 = − ∑ r = 1 ℓ ( z i r + z i r + 1 ) + 2 ∑ k = 1 n z k ∑ r = 1 ℓ H i r k H k i r + 1 H i r i r + 1 . (\log w_P)'=\sum_{r=1}^{\ell}\frac{\dot{H}_{i_ri_{r+1}}}{H_{i_ri_{r+1}}}=-\sum_{r=1}^{\ell}(z_{i_r}+z_{i_{r+1}})+2\sum_{k=1}^{n}z_k\sum_{r=1}^{\ell}\frac{H_{i_rk}H_{ki_{r+1}}}{H_{i_ri_{r+1}}}. ( log w P ) ′ = r = 1 ∑ ℓ H i r i r + 1 H ˙ i r i r + 1 = − r = 1 ∑ ℓ ( z i r + z i r + 1 ) + 2 k = 1 ∑ n z k r = 1 ∑ ℓ H i r i r + 1 H i r k H k i r + 1 . Each vertex occurs at most twice in the first summation, so ∣ ∑ r = 1 ℓ z i r + z i r + 1 ∣ ≤ 2 ∥ z ∥ 1 \left|\sum_{r=1}^{\ell}z_{i_r}+z_{i_{r+1}}\right|\le2\|z\|_1 ∑ r = 1 ℓ z i r + z i r + 1 ≤ 2∥ z ∥ 1 Lemma 5.15 , obtaining
∑ r = 1 ℓ ∣ H i r k H k i r + 1 H i r i r + 1 ∣ = 1 K ∑ r = 1 ℓ d i r d i r + 1 ∣ H i r k H k i r + 1 ∣ ≤ 7 K . \sum_{r=1}^{\ell}\left|\frac{H_{i_rk}H_{ki_{r+1}}}{H_{i_ri_{r+1}}}\right|=\frac{1}{K}\sum_{r=1}^{\ell}\frac{d_{i_r}}{d_{i_{r+1}}}|H_{i_rk}H_{ki_{r+1}}|\le7K. r = 1 ∑ ℓ H i r i r + 1 H i r k H k i r + 1 = K 1 r = 1 ∑ ℓ d i r + 1 d i r ∣ H i r k H k i r + 1 ∣ ≤ 7 K . Consequently, Proposition 5.14 yields, almost everywhere,
∣ ( log d j ) ′ ∣ ≤ ( 2 + 14 K ) ∥ z ∥ 1 ≤ 16 K ∥ z ∥ 1 . |(\log d_j)'|\le(2+14K)\|z\|_1\le16K\|z\|_1. ∣ ( log d j ) ′ ∣ ≤ ( 2 + 14 K ) ∥ z ∥ 1 ≤ 16 K ∥ z ∥ 1 . Since log d j \log d_j log d j [ 0 , 1 ] [0,1] [ 0 , 1 ]
∣ log d j ( 1 ) − log d j ( 0 ) ∣ ≤ 16 K ∥ z ∥ 1 . |\log d_j(1)-\log d_j(0)|\le16K\|z\|_1. ∣ log d j ( 1 ) − log d j ( 0 ) ∣ ≤ 16 K ∥ z ∥ 1 . Using d ( 0 ) = q H d(0)=q_H d ( 0 ) = q H d ( 1 ) = q H ′ d(1)=q_{H'} d ( 1 ) = q H ′ z = log u z=\log u z = log u
Improving the analysis to O ( n 9 / 8 1 + κ ) O(n^{9/8}\sqrt{1+\kappa}) O ( n 9/8 1 + κ )
In this section we provide an improved analysis of the IPM. While the conceptual contribution of Theorem 5.1 was to show that after taking a Newton step from a μ \mu μ x x x μ ′ = μ / ( 1 + δ ) \mu'=\mu/(1+\delta) μ ′ = μ / ( 1 + δ ) x ′ x' x ′ ∥ Q H x ′ , μ ′ ( H x , μ ′ ⋅ ( δ 1 ) ) ∥ 2 2 = O ( δ 2 n ( 1 + κ ) ) \|Q_{H_{x',\mu'}}(H_{x,\mu'}\cdot(\delta\mathbf{1}))\|_2^2=O(\delta^2n(1+\kappa)) ∥ Q H x ′ , μ ′ ( H x , μ ′ ⋅ ( δ 1 )) ∥ 2 2 = O ( δ 2 n ( 1 + κ )) δ = 1 / Ω ( n 5 / 4 1 + κ ) \delta=1/\Omega(n^{5/4}\sqrt{1+\kappa}) δ = 1/Ω ( n 5/4 1 + κ ) O ( n − 3 / 2 ) O(n^{-3/2}) O ( n − 3/2 ) δ = 1 / Ω ( n 9 / 8 1 + κ ) \delta=1/\Omega(n^{9/8}\sqrt{1+\kappa}) δ = 1/Ω ( n 9/8 1 + κ ) O ( n − 3 / 2 ) O(n^{-3/2}) O ( n − 3/2 ) δ 1 \delta\mathbf{1} δ 1 O ( δ 4 n 3 ( 1 + κ ) 2 ) O(\delta^4n^3(1+\kappa)^2) O ( δ 4 n 3 ( 1 + κ ) 2 )
Lemma 6.1 (Centrality error after Newton steps). Let M ∈ P ⋆ ( κ ) M\in\mathcal{P}_\star(\kappa) M ∈ P ⋆ ( κ ) β = 1 4096 n 3 / 2 \beta=\frac{1}{4096n^{3/2}} β = 4096 n 3/2 1 δ = 1 1024 n 9 / 8 1 + κ \delta=\frac{1}{1024n^{9/8}\sqrt{1+\kappa}} δ = 1024 n 9/8 1 + κ 1 ε = β 2 17 n ( 1 + κ ) 2 \varepsilon=\frac{\beta}{2^{17}n(1+\kappa)^2} ε = 2 17 n ( 1 + κ ) 2 β ( x , s , μ ) ∈ N R ( ε ) (x,s,\mu)\in\mathcal{N}^{\mathbb{R}}(\varepsilon) ( x , s , μ ) ∈ N R ( ε ) μ ′ = μ / ( 1 + δ ) \mu'=\mu/(1+\delta) μ ′ = μ / ( 1 + δ ) μ ′ \mu' μ ′ (11) , produce iterates ( x ′ , s ′ ) , ( x ′ ′ , s ′ ′ ) ∈ F > 0 (x',s'),(x'',s'')\in\mathcal{F}_{>0} ( x ′ , s ′ ) , ( x ′′ , s ′′ ) ∈ F > 0 ( x ′ ′ , s ′ ′ , μ ′ ) ∈ N R ( β ) (x'',s'',\mu')\in\mathcal{N}^{\mathbb{R}}(\beta) ( x ′′ , s ′′ , μ ′ ) ∈ N R ( β )
To achieve this result, we need to first prove a few helper lemmas.
Lemma 6.2. Let M ∈ P ⋆ M\in\mathcal{P}_\star M ∈ P ⋆ H = ( I + M ) − 1 H=(I+M)^{-1} H = ( I + M ) − 1 j ∈ [ n ] j\in[n] j ∈ [ n ]
∥ H ∙ j ∥ 2 ∥ H j ∙ ∥ 2 ≤ 2 H j j 1 + κ ^ ( M ) . \|H_{\bullet j}\|_2\|H_{j\bullet}\|_2\le2H_{jj}\sqrt{1+\hat{\kappa}(M)}. ∥ H ∙ j ∥ 2 ∥ H j ∙ ∥ 2 ≤ 2 H j j 1 + κ ^ ( M ) . Proof. Let t ≥ 0 t\ge0 t ≥ 0 [82 Theorem 3.6] . Together with Theorem 1.1, this shows that
∥ ( I + M + t e j e j T ) − 1 ∥ 2 ≤ sup D ∈ D > 0 n ∥ ( I + D ( M + t e j e j T ) D ) − 1 ∥ 2 ≤ 1 + κ ^ ( M + t e j e j T ) ≤ 1 + κ ^ ( M ) . (31) \|(I+M+te_je_j^\mathsf{T})^{-1}\|_2\le\sup_{D\in\mathcal{D}_{>0}^n}\|(I+D(M+te_je_j^\mathsf{T})D)^{-1}\|_2\le\sqrt{1+\hat{\kappa}(M+te_je_j^\mathsf{T})}\le\sqrt{1+\hat{\kappa}(M)}. \tag*{(31)} ∥ ( I + M + t e j e j T ) − 1 ∥ 2 ≤ D ∈ D > 0 n sup ∥ ( I + D ( M + t e j e j T ) D ) − 1 ∥ 2 ≤ 1 + κ ^ ( M + t e j e j T ) ≤ 1 + κ ^ ( M ) . (31) By Sherman-Morrison formula,
( I + M + t e j e j T ) − 1 = ( I + M ) − 1 − t ( I + M ) − 1 e j e j T ( I + M ) − 1 1 + t e j T ( I + M ) − 1 e j = H − t H e j e j T H 1 + t H j j . (I+M+te_je_j^\mathsf{T})^{-1}=(I+M)^{-1}-t\frac{(I+M)^{-1}e_je_j^\mathsf{T}(I+M)^{-1}}{1+te_j^\mathsf{T}(I+M)^{-1}e_j}=H-\frac{tHe_je_j^\mathsf{T}H}{1+tH_{jj}}. ( I + M + t e j e j T ) − 1 = ( I + M ) − 1 − t 1 + t e j T ( I + M ) − 1 e j ( I + M ) − 1 e j e j T ( I + M ) − 1 = H − 1 + t H j j t H e j e j T H . Then, since ∥ H ∙ j ∥ 2 ∥ H j ∙ ∥ 2 = ∥ H ∙ j H j ∙ ∥ 2 = ∥ H e j e j T H ∥ 2 \|H_{\bullet j}\|_2\|H_{j\bullet}\|_2=\|H_{\bullet j}H_{j\bullet}\|_2=\|He_je_j^\mathsf{T}H\|_2 ∥ H ∙ j ∥ 2 ∥ H j ∙ ∥ 2 = ∥ H ∙ j H j ∙ ∥ 2 = ∥ H e j e j T H ∥ 2
∥ H ∙ j ∥ 2 ∥ H j ∙ ∥ 2 ≤ 1 + t H j j t ( ∥ H ∥ 2 + ∥ ( I + M + t e j e j T ) − 1 ∥ 2 ) ≤ 2 ( 1 + t H j j ) t 1 + κ ^ ( M ) , \|H_{\bullet j}\|_2\|H_{j\bullet}\|_2\le\frac{1+tH_{jj}}{t}\left(\|H\|_2+\|(I+M+te_je_j^\mathsf{T})^{-1}\|_2\right)\le\frac{2(1+tH_{jj})}{t}\sqrt{1+\hat{\kappa}(M)}, ∥ H ∙ j ∥ 2 ∥ H j ∙ ∥ 2 ≤ t 1 + t H j j ( ∥ H ∥ 2 + ∥ ( I + M + t e j e j T ) − 1 ∥ 2 ) ≤ t 2 ( 1 + t H j j ) 1 + κ ^ ( M ) , which holds for all t > 0 t>0 t > 0 t → ∞ t\to\infty t → ∞
∥ H ∙ j ∥ 2 ∥ H j ∙ ∥ 2 ≤ inf t > 0 2 1 + κ ^ ( M ) ( 1 + t H j j ) t = 2 H j j 1 + κ ^ ( M ) . \|H_{\bullet j}\|_2\|H_{j\bullet}\|_2\le\inf_{t>0}\frac{2\sqrt{1+\hat{\kappa}(M)}(1+tH_{jj})}{t}=2H_{jj}\sqrt{1+\hat{\kappa}(M)}. ∥ H ∙ j ∥ 2 ∥ H j ∙ ∥ 2 ≤ t > 0 inf t 2 1 + κ ^ ( M ) ( 1 + t H j j ) = 2 H j j 1 + κ ^ ( M ) . :::
Lemma 6.3. Let M ∈ P ⋆ M\in\mathcal{P}_\star M ∈ P ⋆ H = ( I + M ) − 1 H=(I+M)^{-1} H = ( I + M ) − 1 v ∈ R n v\in\mathbb{R}^n v ∈ R n
∥ H diag ( H v ) ∥ 2 ≤ 2 n ( 1 + κ ^ ( M ) ) ∥ v ∥ 2 . \|H\operatorname{diag}(Hv)\|_2\le2\sqrt{n(1+\hat{\kappa}(M))}\|v\|_2. ∥ H diag ( H v ) ∥ 2 ≤ 2 n ( 1 + κ ^ ( M )) ∥ v ∥ 2 . Proof. Since for all i , j ∈ [ n ] i,j\in[n] i , j ∈ [ n ] [ H diag ( H v ) ] i j = ( H v ) j H i j [H\operatorname{diag}(Hv)]_{ij}=(Hv)_jH_{ij} [ H diag ( H v ) ] ij = ( H v ) j H ij Lemma 6.2 we have
∥ H diag ( H v ) ∥ 2 2 ≤ ∑ j = 1 n ∥ H ∙ j ∥ 2 2 ( H v ) j 2 ≤ ∥ v ∥ 2 2 ∑ j = 1 n ∥ H ∙ j ∥ 2 2 ∥ H j ∙ ∥ 2 2 ≤ 4 ( 1 + κ ^ ( M ) ) ∥ v ∥ 2 2 ∑ j = 1 n H j j 2 . \|H\operatorname{diag}(Hv)\|_2^2\le\sum_{j=1}^n\|H_{\bullet j}\|_2^2(Hv)_j^2\le\|v\|_2^2\sum_{j=1}^n\|H_{\bullet j}\|_2^2\|H_{j\bullet}\|_2^2\le4(1+\hat{\kappa}(M))\|v\|_2^2\sum_{j=1}^nH_{jj}^2. ∥ H diag ( H v ) ∥ 2 2 ≤ j = 1 ∑ n ∥ H ∙ j ∥ 2 2 ( H v ) j 2 ≤ ∥ v ∥ 2 2 j = 1 ∑ n ∥ H ∙ j ∥ 2 2 ∥ H j ∙ ∥ 2 2 ≤ 4 ( 1 + κ ^ ( M )) ∥ v ∥ 2 2 j = 1 ∑ n H j j 2 . As a result, the bound for ∥ H diag ( H v ) ∥ 2 \|H\operatorname{diag}(Hv)\|_2 ∥ H diag ( H v ) ∥ 2 H j j ∈ ( 0 , 1 ] H_{jj}\in(0,1] H j j ∈ ( 0 , 1 ] j ∈ [ n ] j\in[n] j ∈ [ n ] Proposition 5.3 and then taking the square root.
Proof of Lemma 6.1. Let M ∈ P ⋆ ( κ ) M\in\mathcal{P}_\star(\kappa) M ∈ P ⋆ ( κ ) ( x , s , μ ) ∈ N R ( ε ) (x,s,\mu)\in\mathcal{N}^{\mathbb{R}}(\varepsilon) ( x , s , μ ) ∈ N R ( ε ) β = 1 4096 n 3 / 2 \beta=\frac{1}{4096n^{3/2}} β = 4096 n 3/2 1 δ = 1 1024 n 9 / 8 1 + κ \delta=\frac{1}{1024n^{9/8}\sqrt{1+\kappa}} δ = 1024 n 9/8 1 + κ 1 μ \mu μ μ ′ = μ / ( 1 + δ ) \mu'=\mu/(1+\delta) μ ′ = μ / ( 1 + δ ) ( x ′ , s ′ ) (x',s') ( x ′ , s ′ ) ( x ′ ′ , s ′ ′ ) (x'',s'') ( x ′′ , s ′′ ) ( x ′ ′ , s ′ ′ , μ ′ ) ∈ N R ( β ) (x'',s'',\mu')\in\mathcal{N}^{\mathbb{R}}(\beta) ( x ′′ , s ′′ , μ ′ ) ∈ N R ( β ) ( x , s , μ ) ∈ N R ( ε ) (x,s,\mu)\in\mathcal{N}^{\mathbb{R}}(\varepsilon) ( x , s , μ ) ∈ N R ( ε ) ∥ Q H x , μ err ( x , s , μ ) ∥ 2 ≤ ε \|Q_{H_{x,\mu}}\operatorname{err}(x,s,\mu)\|_2\le\varepsilon ∥ Q H x , μ err ( x , s , μ ) ∥ 2 ≤ ε ε = β 2 17 n ( 1 + κ ) 2 \varepsilon=\frac{\beta}{2^{17}n(1+\kappa)^2} ε = 2 17 n ( 1 + κ ) 2 β
To simplify notation, let H : = H x , μ ′ H:=H_{x,\mu'} H := H x , μ ′ H ′ : = H x ′ , μ ′ H':=H_{x',\mu'} H ′ := H x ′ , μ ′ H ′ ′ : = H x ′ ′ , μ ′ H'':=H_{x'',\mu'} H ′′ := H x ′′ , μ ′ (4) ,
err ( x , s , μ ′ ) = ( 1 + δ ) err ( x , s , μ ) − δ 1. (32) \operatorname{err}(x,s,\mu')=(1+\delta)\operatorname{err}(x,s,\mu)-\delta\mathbf{1}. \tag*{(32)} err ( x , s , μ ′ ) = ( 1 + δ ) err ( x , s , μ ) − δ 1 . (32) Since α H x , μ ≥ 1 \alpha_{H_{x,\mu}}\ge1 α H x , μ ≥ 1 ∥ err ( x , s , μ ) ∥ 2 ≤ ε \|\operatorname{err}(x,s,\mu)\|_2\le\varepsilon ∥ err ( x , s , μ ) ∥ 2 ≤ ε δ ≤ 1 \delta\le1 δ ≤ 1 2 ε ≤ δ n 2\varepsilon\le\delta\sqrt{n} 2 ε ≤ δ n
∥ err ( x , s , μ ′ ) ∥ 2 ≤ δ n + 2 ∥ err ( x , s , μ ) ∥ 2 ≤ 2 δ n . (33) \|\operatorname{err}(x,s,\mu')\|_2\le\delta\sqrt{n}+2\|\operatorname{err}(x,s,\mu)\|_2\le2\delta\sqrt{n}. \tag*{(33)} ∥ err ( x , s , μ ′ ) ∥ 2 ≤ δ n + 2∥ err ( x , s , μ ) ∥ 2 ≤ 2 δ n . (33) The first Newton step. After taking a single Newton step, the relative change in the iterate is Δ x ˉ : = ( x ′ − x ) / x = Herr ( x , s , μ ′ ) \Delta\bar{x}:=(x'-x)/x=\operatorname{Herr}(x,s,\mu') Δ x ˉ := ( x ′ − x ) / x = Herr ( x , s , μ ′ ) Lemma 5.6 , respectively, we can bound
∥ Δ x ˉ ∥ 2 ≤ 1 + κ ∥ err ( x , s , μ ′ ) ∥ 2 , \|\Delta\bar{x}\|_2 \le\sqrt{1+\kappa}\|\operatorname{err}(x,s,\mu')\|_2, ∥Δ x ˉ ∥ 2 ≤ 1 + κ ∥ err ( x , s , μ ′ ) ∥ 2 , ∥ Q H Δ x ˉ ∥ ∞ ≤ 32 1 + κ ∥ err ( x , s , μ ′ ) ∥ 2 . (34) \|Q_H\Delta\bar{x}\|_\infty\le32\sqrt{1+\kappa}\|\operatorname{err}(x,s,\mu')\|_2. \tag*{(34)} ∥ Q H Δ x ˉ ∥ ∞ ≤ 32 1 + κ ∥ err ( x , s , μ ′ ) ∥ 2 . (34) Additionally, using Lemma 6.3 for v = err ( x , s , μ ′ ) v=\operatorname{err}(x,s,\mu') v = err ( x , s , μ ′ )
∥ H diag ( Δ x ˉ ) ∥ 2 ≤ 2 n ( 1 + κ ) ∥ err ( x , s , μ ′ ) ∥ 2 . (35) \|H\operatorname{diag}(\Delta\bar{x})\|_2 \le2\sqrt{n(1+\kappa)}\|\operatorname{err}(x,s,\mu')\|_2. \tag*{(35)} ∥ H diag ( Δ x ˉ ) ∥ 2 ≤ 2 n ( 1 + κ ) ∥ err ( x , s , μ ′ ) ∥ 2 . (35) The Newton step (11) gives
x ′ s ′ μ ′ = ( 1 n + Δ x ˉ ) ( 1 n − Δ x ˉ ) = 1 n − ( Δ x ˉ ) 2 . (36) \frac{x's'}{\mu'}=(\mathbf{1}_n+\Delta\bar{x})(\mathbf{1}_n-\Delta\bar{x})=\mathbf{1}_n-(\Delta\bar{x})^2. \tag*{(36)} μ ′ x ′ s ′ = ( 1 n + Δ x ˉ ) ( 1 n − Δ x ˉ ) = 1 n − ( Δ x ˉ ) 2 . (36) By (34) and then (33) , ∥ Δ x ˉ ∥ 2 ≤ 2 δ n ( 1 + κ ) ≤ 1 / 512 \|\Delta\bar{x}\|_2\le2\delta\sqrt{n(1+\kappa)}\le1/512 ∥Δ x ˉ ∥ 2 ≤ 2 δ n ( 1 + κ ) ≤ 1/512 x ′ > 0 x'>0 x ′ > 0 (36) , this gives s ′ > 0 s'>0 s ′ > 0 s ′ − s = M ( x ′ − x ) s'-s=M(x'-x) s ′ − s = M ( x ′ − x ) err ( x ′ , s ′ , μ ′ ) = ( Δ x ˉ ) 2 \operatorname{err}(x',s',\mu')=(\Delta\bar{x})^2 err ( x ′ , s ′ , μ ′ ) = ( Δ x ˉ ) 2
The second Newton step. The second relative change is Δ x ˉ ′ ′ : = ( x ′ ′ − x ′ ) / x ′ = H ′ ( Δ x ˉ ) 2 \Delta\bar{x}'':=(x''-x')/x'=H'(\Delta\bar{x})^2 Δ x ˉ ′′ := ( x ′′ − x ′ ) / x ′ = H ′ ( Δ x ˉ ) 2 err ( x ′ ′ , s ′ ′ , μ ′ ) = ( Δ x ˉ ) 2 \operatorname{err}(x'',s'',\mu')=(\Delta\bar{x})^2 err ( x ′′ , s ′′ , μ ′ ) = ( Δ x ˉ ) 2 Lemma 5.10 , we bound the Rump norm of centrality error by a product of two norms:
∥ Q H ′ err ( x ′ ′ , s ′ ′ , μ ′ ) ∥ 2 = ∥ Q H ′ ( Δ x ˉ ) 2 ∥ 2 ≤ ∥ Q H ′ Δ x ˉ ∥ ∞ ∥ Δ x ˉ ∥ 2 . (37) \|Q_{H'}\operatorname{err}(x'',s'',\mu')\|_2=\|Q_{H'}(\Delta\bar{x})^2\|_2\le\|Q_{H'}\Delta\bar{x}\|_\infty\|\Delta\bar{x}\|_2. \tag*{(37)} ∥ Q H ′ err ( x ′′ , s ′′ , μ ′ ) ∥ 2 = ∥ Q H ′ ( Δ x ˉ ) 2 ∥ 2 ≤ ∥ Q H ′ Δ x ˉ ∥ ∞ ∥Δ x ˉ ∥ 2 . (37) We will bound these norms using (34) and (35) . To do so, we first account for the changes from H H H H ′ H' H ′ q H q_H q H q H ′ q_{H'} q H ′ U : = diag ( 1 + Δ x ˉ ) U:=\operatorname{diag}(1+\Delta\bar{x}) U := diag ( 1 + Δ x ˉ )
H ′ = U − 1 ( U − 2 − I n + H − 1 ) − 1 U − 1 = U − 1 ( I n + H ( U − 2 − I n ) ) − 1 H U − 1 . (38) H'=U^{-1}(U^{-2}-I_n+H^{-1})^{-1}U^{-1}=U^{-1}(I_n+H(U^{-2}-I_n))^{-1}HU^{-1}. \tag*{(38)} H ′ = U − 1 ( U − 2 − I n + H − 1 ) − 1 U − 1 = U − 1 ( I n + H ( U − 2 − I n ) ) − 1 H U − 1 . (38) Since ∣ Δ x ˉ i ∣ ≤ 1 / 4 |\Delta\bar{x}_i|\le1/4 ∣Δ x ˉ i ∣ ≤ 1/4
( U − 2 − I n ) i i = − Δ x ˉ i 2 + Δ x ˉ i ( 1 + Δ x ˉ i ) 2 (U^{-2}-I_n)_{ii}=-\Delta\bar{x}_i\frac{2+\Delta\bar{x}_i}{(1+\Delta\bar{x}_i)^2} ( U − 2 − I n ) ii = − Δ x ˉ i ( 1 + Δ x ˉ i ) 2 2 + Δ x ˉ i and (35) imply
∥ H ( U − 2 − I n ) ∥ 2 ≤ 4 ∥ H diag ( Δ x ˉ ) ∥ 2 ≤ 8 n ( 1 + κ ) ∥ err ( x , s , μ ′ ) ∥ 2 ≤ 16 δ n 1 + κ ≤ 1 2 , \|H(U^{-2}-I_n)\|_2\le4\|H\operatorname{diag}(\Delta\bar{x})\|_2\le8\sqrt{n(1+\kappa)}\|\operatorname{err}(x,s,\mu')\|_2\le16\delta n\sqrt{1+\kappa}\le\frac{1}{2}, ∥ H ( U − 2 − I n ) ∥ 2 ≤ 4∥ H diag ( Δ x ˉ ) ∥ 2 ≤ 8 n ( 1 + κ ) ∥ err ( x , s , μ ′ ) ∥ 2 ≤ 16 δ n 1 + κ ≤ 2 1 , where we bound ∥ err ( x , s , μ ′ ) ∥ 2 \|\operatorname{err}(x,s,\mu')\|_2 ∥ err ( x , s , μ ′ ) ∥ 2 (33) . Hence ∥ ( I n + H ( U − 2 − I n ) ) − 1 ∥ 2 ≤ 2 \|(I_n+H(U^{-2}-I_n))^{-1}\|_2\le2 ∥ ( I n + H ( U − 2 − I n ) ) − 1 ∥ 2 ≤ 2 ∥ U − 1 ∥ 2 ≤ 4 / 3 \|U^{-1}\|_2\le4/3 ∥ U − 1 ∥ 2 ≤ 4/3 U − 1 U^{-1} U − 1 diag ( Δ x ˉ ) \operatorname{diag}(\Delta\bar{x}) diag ( Δ x ˉ ) (38) gives
∥ H ′ diag ( Δ x ˉ ) ∥ 2 ≤ 4 ∥ H diag ( Δ x ˉ ) ∥ 2 ≤ 8 n ( 1 + κ ) ∥ err ( x , s , μ ′ ) ∥ 2 . (39) \|H'\operatorname{diag}(\Delta\bar{x})\|_2\le4\|H\operatorname{diag}(\Delta\bar{x})\|_2\le8\sqrt{n(1+\kappa)}\|\operatorname{err}(x,s,\mu')\|_2. \tag*{(39)} ∥ H ′ diag ( Δ x ˉ ) ∥ 2 ≤ 4∥ H diag ( Δ x ˉ ) ∥ 2 ≤ 8 n ( 1 + κ ) ∥ err ( x , s , μ ′ ) ∥ 2 . (39) For the weights, using Lemma 5.7 , ∣ log ( 1 + a ) ∣ ≤ 2 ∣ a ∣ |\log(1+a)|\le2|a| ∣ log ( 1 + a ) ∣ ≤ 2∣ a ∣ ∣ a ∣ ≤ 1 / 2 |a|\le1/2 ∣ a ∣ ≤ 1/2 ∥ Δ x ˉ ∥ 2 ≤ 2 δ n ( 1 + κ ) \|\Delta\bar{x}\|_2\le2\delta\sqrt{n(1+\kappa)} ∥Δ x ˉ ∥ 2 ≤ 2 δ n ( 1 + κ ) (34) give
∥ log q H ′ − log q H ∥ ∞ ≤ 32 K ∥ Δ x ˉ ∥ 1 ≤ 32 K n 1 / 2 ∥ Δ x ˉ ∥ 2 ≤ 64 δ K n 1 + κ ≤ 1 2 , (40) \|\log q_{H'}-\log q_H\|_\infty\le32K\|\Delta\bar{x}\|_1\le32Kn^{1/2}\|\Delta\bar{x}\|_2\le64\delta Kn\sqrt{1+\kappa}\le\frac{1}{2}, \tag*{(40)} ∥ log q H ′ − log q H ∥ ∞ ≤ 32 K ∥Δ x ˉ ∥ 1 ≤ 32 K n 1/2 ∥Δ x ˉ ∥ 2 ≤ 64 δ K n 1 + κ ≤ 2 1 , (40) and consequently, q H ′ / q H ≤ e 1 / 2 ≤ 2 q_{H'}/q_H\le e^{1/2}\le2 q H ′ / q H ≤ e 1/2 ≤ 2
We can now bound the two norms. From (39) and (34) ,
∥ Δ x ˉ ′ ∥ 2 = ∥ H ′ ( Δ x ˉ ) 2 ∥ 2 ≤ ∥ H ′ diag ( Δ x ˉ ) ∥ 2 ∥ Δ x ˉ ∥ 2 ≤ 8 n ( 1 + κ ) ∥ err ( x , s , μ ′ ) ∥ 2 2 . (41) \|\Delta\bar{x}'\|_2=\|H'(\Delta\bar{x})^2\|_2\le\|H'\operatorname{diag}(\Delta\bar{x})\|_2\|\Delta\bar{x}\|_2\le8\sqrt{n(1+\kappa)}\|\operatorname{err}(x,s,\mu')\|_2^2. \tag*{(41)} ∥Δ x ˉ ′ ∥ 2 = ∥ H ′ ( Δ x ˉ ) 2 ∥ 2 ≤ ∥ H ′ diag ( Δ x ˉ ) ∥ 2 ∥Δ x ˉ ∥ 2 ≤ 8 n ( 1 + κ ) ∥ err ( x , s , μ ′ ) ∥ 2 2 . (41) For the other norm, Lemma 5.4 gives ∥ Q H ′ H ′ Q H ′ − 1 ∥ max ≤ K \|Q_{H'}H'Q_{H'}^{-1}\|_{\max}\le K ∥ Q H ′ H ′ Q H ′ − 1 ∥ m a x ≤ K (40) , this yields
∥ Q H ′ Δ x ˉ ′ ∥ ∞ = ∥ Q H ′ H ′ ( Δ x ˉ ) 2 ∥ ∞ ≤ K ∥ Q H ′ ( Δ x ˉ ) 2 ∥ 1 ≤ 2 K ∥ Q H Δ x ˉ ∥ ∞ ∥ Δ x ˉ ∥ 1 ≤ 512 n ( 1 + κ ) ∥ err ( x , s , μ ′ ) ∥ 2 2 , (42) \|Q_{H'}\Delta\bar{x}'\|_\infty=\|Q_{H'}H'(\Delta\bar{x})^2\|_\infty\le K\|Q_{H'}(\Delta\bar{x})^2\|_1\le2K\|Q_H\Delta\bar{x}\|_\infty\|\Delta\bar{x}\|_1\le512\sqrt{n(1+\kappa)}\|\operatorname{err}(x,s,\mu')\|_2^2, \tag*{(42)} ∥ Q H ′ Δ x ˉ ′ ∥ ∞ = ∥ Q H ′ H ′ ( Δ x ˉ ) 2 ∥ ∞ ≤ K ∥ Q H ′ ( Δ x ˉ ) 2 ∥ 1 ≤ 2 K ∥ Q H Δ x ˉ ∥ ∞ ∥Δ x ˉ ∥ 1 ≤ 512 n ( 1 + κ ) ∥ err ( x , s , μ ′ ) ∥ 2 2 , (42) where the last inequality uses (34) and ∥ Δ x ˉ ∥ 1 ≤ n ∥ Δ x ˉ ∥ 2 \|\Delta\bar{x}\|_1\le\sqrt{n}\|\Delta\bar{x}\|_2 ∥Δ x ˉ ∥ 1 ≤ n ∥Δ x ˉ ∥ 2
Bounding the centrality error. Combining (41) and (42) in (37) gives
∥ Q H ′ err ( x ′ ′ , s ′ ′ , μ ′ ) ∥ 2 ≤ 4096 n ( 1 + κ ) 2 ∥ err ( x , s , μ ′ ) ∥ 2 4 . (43) \lVert Q_{H'}\operatorname{err}(x^{\prime\prime},s^{\prime\prime},\mu^{\prime})\rVert_2 \le4096n(1+\kappa)^2\lVert\operatorname{err}(x,s,\mu^{\prime})\rVert_2^4. \tag*{(43)} ∥ Q H ′ err ( x ′′ , s ′′ , μ ′ ) ∥ 2 ≤ 4096 n ( 1 + κ ) 2 ∥ err ( x , s , μ ′ ) ∥ 2 4 . (43) To replace Q H ′ Q_{H'} Q H ′ Q H ′ ′ Q_{H^{\prime\prime}} Q H ′′ (41) and (33) give
∥ Δ x ˉ ′ ∥ 1 ≤ n ∥ Δ x ˉ ′ ∥ 2 ≤ 8 n ( 1 + κ ) ∥ err ( x , s , μ ′ ) ∥ 2 2 ≤ 8 512 2 ≤ 1 512 . \lVert\Delta\bar{x}^{\prime}\rVert_1 \le\sqrt{n}\lVert\Delta\bar{x}^{\prime}\rVert_2 \le8n(1+\kappa)\lVert\operatorname{err}(x,s,\mu^{\prime})\rVert_2^2 \le\frac{8}{512^2} \le\frac{1}{512}. ∥ Δ x ˉ ′ ∥ 1 ≤ n ∥ Δ x ˉ ′ ∥ 2 ≤ 8 n ( 1 + κ ) ∥ err ( x , s , μ ′ ) ∥ 2 2 ≤ 51 2 2 8 ≤ 512 1 . Thus x ′ ′ , s ′ ′ > 0 x^{\prime\prime},s^{\prime\prime}>0 x ′′ , s ′′ > 0 (36) to the second step. The same application of Lemma 5.7 as in (40) gives q H ′ ′ ≤ 2 q H ′ q_{H^{\prime\prime}}\le2q_{H'} q H ′′ ≤ 2 q H ′
∥ Q H ′ ′ err ( x ′ ′ , s ′ ′ , μ ′ ) ∥ 2 ≤ 8192 n ( 1 + κ ) 2 ∥ err ( x , s , μ ′ ) ∥ 2 4 ≤ 8192 δ 4 n 3 ( 1 + κ ) 2 + 2 16 n ( 1 + κ ) 2 ∥ err ( x , s , μ ) ∥ 2 2 ≤ β 2 15 + β 2 ≤ β . (44) \begin{aligned}
\lVert Q_{H^{\prime\prime}}\operatorname{err}(x^{\prime\prime},s^{\prime\prime},\mu^{\prime})\rVert_2
&\le8192n(1+\kappa)^2\lVert\operatorname{err}(x,s,\mu^{\prime})\rVert_2^4\\
&\le8192\delta^4n^3(1+\kappa)^2+2^{16}n(1+\kappa)^2\lVert\operatorname{err}(x,s,\mu)\rVert_2^2\\
&\le\frac{\beta}{2^{15}}+\frac{\beta}{2}\le\beta. \tag*{(44)}
\end{aligned} ∥ Q H ′′ err ( x ′′ , s ′′ , μ ′ ) ∥ 2 ≤ 8192 n ( 1 + κ ) 2 ∥ err ( x , s , μ ′ ) ∥ 2 4 ≤ 8192 δ 4 n 3 ( 1 + κ ) 2 + 2 16 n ( 1 + κ ) 2 ∥ err ( x , s , μ ) ∥ 2 2 ≤ 2 15 β + 2 β ≤ β . (44) For the second inequality, we apply to (33) the bound ( a + b ) 4 ≤ a 4 + 4 b (a+b)^4\le a^4+4b ( a + b ) 4 ≤ a 4 + 4 b 0 ≤ a ≤ a + b ≤ 1 0\le a\le a+b\le1 0 ≤ a ≤ a + b ≤ 1 a = δ n a=\delta\sqrt{n} a = δ n b = 2 ∥ err ( x , s , μ ) ∥ 2 2 b=2\lVert\operatorname{err}(x,s,\mu)\rVert_2^2 b = 2 ∥ err ( x , s , μ ) ∥ 2 2 a + b ≤ 1 a+b\le1 a + b ≤ 1 (33) . The last line uses the choice of δ \delta δ ∥ err ( x , s , μ ) ∥ 2 ≤ ε \lVert\operatorname{err}(x,s,\mu)\rVert_2\le\varepsilon ∥ err ( x , s , μ ) ∥ 2 ≤ ε ( x ′ ′ , s ′ ′ , μ ′ ) ∈ N R ( β ) (x^{\prime\prime},s^{\prime\prime},\mu^{\prime})\in\mathcal{N}^{R}(\beta) ( x ′′ , s ′′ , μ ′ ) ∈ N R ( β )
Proposition 5.8 reduces the Rump norm error from β = O ( n − 3 / 2 ) \beta=O(n^{-3/2}) β = O ( n − 3/2 ) ε \varepsilon ε O ( log log ( n + κ ) ) O(\log\log(n+\kappa)) O ( log log ( n + κ )) N R ( β ) \mathcal{N}^{R}(\beta) N R ( β ) μ ′ \mu^{\prime} μ ′ μ \mu μ O ( n 9 / 8 1 + κ ) O(n^{9/8}\sqrt{1+\kappa}) O ( n 9/8 1 + κ ) O ( n 9 / 8 1 + κ log log ( n + κ ) ) O(n^{9/8}\sqrt{1+\kappa}\log\log(n+\kappa)) O ( n 9/8 1 + κ log log ( n + κ ))
The extra log log ( n + κ ) \log\log(n+\kappa) log log ( n + κ ) O ( n − 3 / 2 ) O(n^{-3/2}) O ( n − 3/2 ) 4
Theorem 6.4. Let M ∈ P ⋆ ( κ ) M\in\mathcal{P}_{\star}(\kappa) M ∈ P ⋆ ( κ ) β ∈ [ 1 8192 n 3 / 2 , 1 4096 n 3 / 2 ] \beta\in\left[\frac{1}{8192n^{3/2}},\frac{1}{4096n^{3/2}}\right] β ∈ [ 8192 n 3/2 1 , 4096 n 3/2 1 ] δ ∈ [ 1 2436 n 9 / 8 1 + κ , 1 1218 n 9 / 8 1 + κ ] \delta\in\left[\frac{1}{2436n^{9/8}\sqrt{1+\kappa}},\frac{1}{1218n^{9/8}\sqrt{1+\kappa}}\right] δ ∈ [ 2436 n 9/8 1 + κ 1 , 1218 n 9/8 1 + κ 1 ] ( x , s , μ ) ∈ N R ( β ) (x,s,\mu)\in\mathcal{N}^{R}(\beta) ( x , s , μ ) ∈ N R ( β ) μ ′ = μ / ( 1 + δ ) \mu^{\prime}=\mu/(1+\delta) μ ′ = μ / ( 1 + δ ) μ ′ \mu^{\prime} μ ′ (11) , produce two iterates ( x ′ , s ′ ) , ( x ′ ′ , s ′ ′ ) ∈ F > 0 (x^{\prime},s^{\prime}),(x^{\prime\prime},s^{\prime\prime})\in\mathcal{F}_{>0} ( x ′ , s ′ ) , ( x ′′ , s ′′ ) ∈ F > 0 ( x ′ ′ , s ′ ′ , μ ′ ) ∈ N R ( β ) (x^{\prime\prime},s^{\prime\prime},\mu^{\prime})\in\mathcal{N}^{R}(\beta) ( x ′′ , s ′′ , μ ′ ) ∈ N R ( β )
Hardness results
Hardness of testing membership between P \mathcal{P} P P 0 \mathcal{P}_0 P 0
In this section, we prove Theorem 1.3. Our main tool is based on the characterization of P \mathcal{P} P [66 ] . For convenience, we restate the definition of the sign-real spectral radius of a matrix:
Definition 7.1 (Sign-real spectral radius restated). Let S n \mathcal{S}^n S n ± 1 \pm1 ± 1 M ∈ R n × n M\in\mathbb{R}^{n\times n} M ∈ R n × n sign-real spectral radius is defined as
ρ 0 s ( M ) : = max S ∈ S n { ∣ λ ∣ : S M x = λ x , for some x ∈ R n ∖ { 0 } } . \rho_0^s(M):=\max_{S\in\mathcal{S}^n}\{|\lambda|:SMx=\lambda x,\text{ for some }x\in\mathbb{R}^n\setminus\{0\}\}. ρ 0 s ( M ) := S ∈ S n max { ∣ λ ∣ : S M x = λ x , for some x ∈ R n ∖ { 0 }} . with the convention that ρ 0 s ( M ) : = 0 \rho_0^s(M):=0 ρ 0 s ( M ) := 0 S M SM S M S ∈ S n S\in\mathcal{S}^n S ∈ S n
It follows directly from the definition that ρ 0 s ( α M ) = ∣ α ∣ ρ 0 s ( M ) \rho_0^s(\alpha M)=|\alpha|\rho_0^s(M) ρ 0 s ( α M ) = ∣ α ∣ ρ 0 s ( M ) α ∈ R \alpha\in\mathbb{R} α ∈ R [66 ] for further properties of ρ 0 s ( M ) \rho_0^s(M) ρ 0 s ( M )
Definition 7.2 (Cayley transform). For M ∈ R n × n M\in\mathbb{R}^{n\times n} M ∈ R n × n I n − M I_n-M I n − M M M M C ( M ) : = ( I n − M ) − 1 ( I n + M ) \mathcal{C}(M):=(I_n-M)^{-1}(I_n+M) C ( M ) := ( I n − M ) − 1 ( I n + M )
Whenever a Cayley transform of M M M I n + M I_n+M I n + M
Proposition 7.3 (Inverse Cayley transform). For M ∈ R n × n M \in\mathbb{R}^{n \times n} M ∈ R n × n I n + M I_n+M I n + M C − 1 ( M ) : = ( M − I n ) ( M + I n ) − 1 C^{-1}(M) := (M-I_n)(M+I_n)^{-1} C − 1 ( M ) := ( M − I n ) ( M + I n ) − 1 C ( C − 1 ( M ) ) = M C(C^{-1}(M))=M C ( C − 1 ( M )) = M I n − M I_n-M I n − M C ( M ) C(M) C ( M ) C − 1 ( C ( M ) ) = M C^{-1}(C(M))=M C − 1 ( C ( M )) = M
As a main result in [66 ] that relates the sign-real spectral radius to the matrix classes considered in LCPs, Rump showed that ρ 0 S ( M ) < 1 \rho_0^S(M)<1 ρ 0 S ( M ) < 1 P P P
Theorem 7.4 (Rump [66 ] ). For M ∈ R n × n M \in\mathbb{R}^{n \times n} M ∈ R n × n I n − M I_n-M I n − M ρ 0 S ( M ) < 1 \rho_0^S(M)<1 ρ 0 S ( M ) < 1 C ( M ) ∈ P C(M) \in\mathcal{P} C ( M ) ∈ P
We refer to [68 ] for a self-contained proof of this theorem. For a rational matrix M ∈ Q n × n M \in\mathbb{Q}^{n \times n} M ∈ Q n × n P P P ρ 0 S ( M ) \rho_0^S(M) ρ 0 S ( M ) 1 1 1 Theorem 7.4 , that is, there exists R > 0 R>0 R > 0 ρ 0 S ( M ) < 1 \rho_0^S(M)<1 ρ 0 S ( M ) < 1 ρ 0 S ( M ) ≤ 1 − 1 / R \rho_0^S(M)\leq1-1/R ρ 0 S ( M ) ≤ 1 − 1/ R R R R M M M
Lemma 7.5. For any M ∈ Q n × n M \in\mathbb{Q}^{n \times n} M ∈ Q n × n R : = C n ( 1 + n ∥ M ∥ max ) n − 1 R := C^n(1+n\lVert M\rVert_{\max})^{n-1} R := C n ( 1 + n ∥ M ∥ m a x ) n − 1 C C C M M M ρ 0 S ( M ) < 1 \rho_0^S(M)<1 ρ 0 S ( M ) < 1 ρ 0 S ( M ) ≤ 1 − 1 R \rho_0^S(M)\leq1-\frac{1}{R} ρ 0 S ( M ) ≤ 1 − R 1 log R = O ( poly ( n , length ( M ) ) ) \log R=O(\operatorname{poly}(n,\operatorname{length}(M))) log R = O ( poly ( n , length ( M )))
The proof is similar to bounding the maximum absolute value of eigenvalues by the encoding length of the matrix. We defer it to § 9.3 . By Lemma 2.4, any P 0 P_0 P 0 P P P P 0 = cl ( P ) P_0=\operatorname{cl}(P) P 0 = cl ( P ) ρ 0 S ( . ) \rho_0^S(.) ρ 0 S ( . ) Theorem 7.4 to P 0 P_0 P 0
Theorem 7.6 (Closed Rump’s theorem). For M ∈ R n × n M \in\mathbb{R}^{n \times n} M ∈ R n × n I n − M I_n-M I n − M ρ 0 S ( M ) ≤ 1 \rho_0^S(M)\leq1 ρ 0 S ( M ) ≤ 1 C ( M ) ∈ P 0 C(M)\in P_0 C ( M ) ∈ P 0
The proof can be found in § 9.3 . Following the road-map described in § 3.4 , we provide Algorithm 2 for the detailed construction of a matrix M ~ \widetilde{M} M M M M M ∈ P ∩ Q n × n M\in\mathcal{P}\cap\mathbb{Q}^{n\times n} M ∈ P ∩ Q n × n M ~ ∈ P ~ ∩ Q n × n \widetilde{M}\in\widetilde{\mathcal{P}}\cap\mathbb{Q}^{n\times n} M ∈ P ∩ Q n × n P ⊆ P ~ ⊆ P 0 \mathcal{P}\subseteq\widetilde{\mathcal{P}}\subseteq P_0 P ⊆ P ⊆ P 0 Theorem 1.3 , using Theorem 7.4 , Lemma 7.5 and Theorem 7.6.
Figure 2.
Lemma 7.7. Given M ∈ Q n × n M\in\mathbb{Q}^{n\times n} M ∈ Q n × n P ~ \widetilde{\mathcal{P}} P P ⊆ P ~ ⊆ P 0 \mathcal{P}\subseteq\widetilde{\mathcal{P}}\subseteq P_0 P ⊆ P ⊆ P 0 M ~ ∈ Q n × n \widetilde{M}\in\mathbb{Q}^{n\times n} M ∈ Q n × n M ∈ P M\in\mathcal{P} M ∈ P M ~ ∈ P ~ \widetilde{M}\in\widetilde{\mathcal{P}} M ∈ P
Proof. Using det ( I n + M ) = ∑ I ⊆ [ n ] det ( M I , I ) \det(I_n+M)=\sum_{I\subseteq[n]}\det(M_{I,I}) det ( I n + M ) = ∑ I ⊆ [ n ] det ( M I , I ) det ( I n + M ) ≤ 0 \det(I_n+M)\leq0 det ( I n + M ) ≤ 0 M ∉ P M\notin\mathcal{P} M ∈ / P M ~ = − I n ∉ P ~ \widetilde{M}=-I_n\notin\widetilde{\mathcal{P}} M = − I n ∈ / P N = ( M − I n ) ( M + I n ) − 1 N=(M-I_n)(M+I_n)^{-1} N = ( M − I n ) ( M + I n ) − 1 C ( N ) = M C(N)=M C ( N ) = M Proposition 7.3 . R R R ρ 0 S ( N ) < 1 \rho_0^S(N)<1 ρ 0 S ( N ) < 1 ρ 0 S ( N ) ≤ 1 − 1 R \rho_0^S(N)\leq1-\frac{1}{R} ρ 0 S ( N ) ≤ 1 − R 1 Theorem 7.4 and Lemma 7.5, M ∈ P M\in\mathcal{P} M ∈ P ρ 0 S ( N ) ≤ 1 − 1 R \rho_0^S(N)\leq1-\frac{1}{R} ρ 0 S ( N ) ≤ 1 − R 1 α = 1 + 1 2 R \alpha=1+\frac{1}{2R} α = 1 + 2 R 1
Claim 7.8. If M ∈ P M\in\mathcal{P} M ∈ P ρ 0 S ( α N ) < 1 \rho_0^S(\alpha N)<1 ρ 0 S ( α N ) < 1 M ∉ P M\notin\mathcal{P} M ∈ / P ρ 0 S ( α N ) > 1 \rho_0^S(\alpha N)>1 ρ 0 S ( α N ) > 1
Proof. If M ∈ P M \in\mathcal{P} M ∈ P ρ 0 S ( N ) ≤ 1 − 1 R \rho_0^S(N) \le1-\frac{1}{R} ρ 0 S ( N ) ≤ 1 − R 1 ρ 0 S ( α N ) = α ρ 0 S ( N ) ≤ ( 1 + 1 2 R ) ( 1 − 1 R ) < 1 \rho_0^S(\alpha N)=\alpha\rho_0^S(N)\le\left(1+\frac{1}{2R}\right)\left(1-\frac{1}{R}\right)<1 ρ 0 S ( α N ) = α ρ 0 S ( N ) ≤ ( 1 + 2 R 1 ) ( 1 − R 1 ) < 1 M ∉ P M\notin\mathcal{P} M ∈ / P ρ 0 S ( N ) ≥ 1 \rho_0^S(N)\ge1 ρ 0 S ( N ) ≥ 1 ρ 0 S ( α N ) = α ρ 0 S ( N ) ≥ α > 1 \rho_0^S(\alpha N)=\alpha\rho_0^S(N)\ge\alpha>1 ρ 0 S ( α N ) = α ρ 0 S ( N ) ≥ α > 1
Claim 7.9. If det ( I n − α N ) = 0 \det(I_n-\alpha N)=0 det ( I n − α N ) = 0 M ∉ P M\notin\mathcal{P} M ∈ / P
Proof. Suppose that M ∈ P M\in\mathcal{P} M ∈ P det ( I n − α N ) = 0 \det(I_n-\alpha N)=0 det ( I n − α N ) = 0 x ≠ 0 n x\ne0_n x = 0 n α N x = x \alpha Nx=x α N x = x ρ 0 S ( α N ) \rho_0^S(\alpha N) ρ 0 S ( α N ) ρ 0 S ( α N ) ≥ 1 \rho_0^S(\alpha N)\ge1 ρ 0 S ( α N ) ≥ 1 Claim 7.8 .
In case det ( I n − α N ) = 0 \det(I_n-\alpha N)=0 det ( I n − α N ) = 0 M ~ = − I n ∉ P ~ \widetilde{M}=-I_n\notin\widetilde{\mathcal{P}} M = − I n ∈ / P M ~ = ( I n − α N ) − 1 ( I n + α N ) = C ( α N ) \widetilde{M}=(I_n-\alpha N)^{-1}(I_n+\alpha N)=C(\alpha N) M = ( I n − α N ) − 1 ( I n + α N ) = C ( α N ) M ∈ P M\in\mathcal{P} M ∈ P M ~ ∈ P ~ \widetilde{M}\in\widetilde{\mathcal{P}} M ∈ P M ∈ P M\in\mathcal{P} M ∈ P ρ 0 S ( α N ) < 1 \rho_0^S(\alpha N)<1 ρ 0 S ( α N ) < 1 Claim 7.8 , then M ~ ∈ P \widetilde{M}\in\mathcal{P} M ∈ P Theorem 7.4 . Since P ⊆ P ~ \mathcal{P}\subseteq\widetilde{\mathcal{P}} P ⊆ P M ~ ∈ P ~ \widetilde{M}\in\widetilde{\mathcal{P}} M ∈ P M ∉ P M\notin\mathcal{P} M ∈ / P ρ 0 S ( α N ) > 1 \rho_0^S(\alpha N)>1 ρ 0 S ( α N ) > 1 Claim 7.8 . Using Theorem 7.6, M ~ ∉ P 0 \widetilde{M}\notin\mathcal{P}_0 M ∈ / P 0 P ~ ⊆ P 0 \widetilde{\mathcal{P}}\subseteq\mathcal{P}_0 P ⊆ P 0 M ~ ∉ P ~ \widetilde{M}\notin\widetilde{\mathcal{P}} M ∈ / P
Since log R = O ( poly ( n , length ( M ) ) ) \log R=O(\operatorname{poly}(n,\operatorname{length}(M))) log R = O ( poly ( n , length ( M ))) length ( N ) = O ( poly ( n , length ( M ) ) ) \operatorname{length}(N)=O(\operatorname{poly}(n,\operatorname{length}(M))) length ( N ) = O ( poly ( n , length ( M ))) α \alpha α M ~ \widetilde{M} M n n n length ( M ) \operatorname{length}(M) length ( M )
Proof of Theorem 1.3 . The first statement follows from Lemma 7.7 and that deciding if M ∈ Q n × n M\in\mathbb{Q}^{n\times n} M ∈ Q n × n P \mathcal{P} P Theorem 2.3 , a P ⋆ \mathcal{P}_\star P ⋆ [77 ] , thus the second statement follows.
Hardness of handicap number computation
In this section, we prove Theorem 1.4 : for any graph G = ( V , E ) G=(V,E) G = ( V , E ) P \mathcal{P} P G G G A G A_G A G G G G G G G A G ∈ { 0 , ± 1 } ∣ V ∣ × ∣ E ∣ A_G\in\{0,\pm1\}^{|V|\times|E|} A G ∈ { 0 , ± 1 } ∣ V ∣ × ∣ E ∣ e = ( i , j ) ∈ E e=(i,j)\in E e = ( i , j ) ∈ E e e e A G A_G A G + 1 +1 + 1 i i i − 1 -1 − 1 j j j 0 0 0 L G : = A G A G T ∈ R n × n L_G:=A_GA_G^{\mathsf{T}}\in\mathbb{R}^{n\times n} L G := A G A G T ∈ R n × n G G G [26 ] .
Proposition 7.10. For a graph G = ( V , E ) G=(V,E) G = ( V , E )
maxcut ( G ) = 1 4 max v ∈ { − 1 , 1 } n v T L G v . \operatorname{maxcut}(G)=\frac{1}{4}\max_{v\in\{-1,1\}^{n}}v^{\mathsf{T}}L_Gv. maxcut ( G ) = 4 1 v ∈ { − 1 , 1 } n max v T L G v . When M ∈ P ⋆ M\in\mathcal{P}_\star M ∈ P ⋆ x T M x x^{\mathsf{T}}Mx x T M x x ∈ R n x\in\mathbb{R}^n x ∈ R n x x x M x Mx M x x T M x x^{\mathsf{T}}Mx x T M x § 9.3 .
Proposition 7.11. Let M ∈ P ⋆ M\in\mathcal{P}_\star M ∈ P ⋆ x ∈ R n x\in\mathbb{R}^n x ∈ R n y = 1 2 ( x + M x ) y=\frac{1}{2}(x+Mx) y = 2 1 ( x + M x ) N = ( M − I n ) / 2 N=(M-I_n)/2 N = ( M − I n ) /2
x T M x = ∥ y ∥ 2 − ∥ N ( I n + N ) − 1 y ∥ 2 . x^{\mathsf{T}}Mx=\lVert y\rVert^2-\lVert N(I_n+N)^{-1}y\rVert^2. x T M x = ∥ y ∥ 2 − ∥ N ( I n + N ) − 1 y ∥ 2 . If N 2 = 0 n × n N^2=0_{n\times n} N 2 = 0 n × n x T M x = y T ( I n − N T N ) y x^{\mathsf{T}}Mx=y^{\mathsf{T}}(I_n-N^{\mathsf{T}}N)y x T M x = y T ( I n − N T N ) y
We derive the following corollary from Proposition 7.10; the proof is also given in § 9.3 .
Proposition 7.12. Let G = ( V , E ) G=(V,E) G = ( V , E ) ∣ V ∣ = n ≥ 2 |V|=n\ge2 ∣ V ∣ = n ≥ 2 ∣ E ∣ = m |E|=m ∣ E ∣ = m v ∈ R n v\in\mathbb{R}^n v ∈ R n
v T ( L G − I n ) v ≤ ∥ v ∥ ∞ 2 ( 4 maxcut ( G ) − n ) . v^{\mathsf{T}}(L_G-I_n)v\le\lVert v\rVert_\infty^2(4\operatorname{maxcut}(G)-n). v T ( L G − I n ) v ≤ ∥ v ∥ ∞ 2 ( 4 maxcut ( G ) − n ) . We now introduce our construction of M G M_G M G
Lemma 7.13. Let G = ( V , E ) G=(V,E) G = ( V , E ) ∣ V ∣ = n ≥ 2 |V|=n\ge2 ∣ V ∣ = n ≥ 2 ∣ E ∣ = m |E|=m ∣ E ∣ = m A G ∈ R n × m A_G\in\mathbb{R}^{n\times m} A G ∈ R n × m
M G : = ( I m 2 A G T 0 0 I n 21 n 0 0 1 ) ∈ Z ( m + n + 1 ) × ( m + n + 1 ) . (45) M_G:=\begin{pmatrix}
I_m & 2A_G^{\mathsf{T}} & 0\\
0 & I_n & 2\mathbf{1}_n\\
0 & 0 & 1
\end{pmatrix}\in\mathbb{Z}^{(m+n+1)\times(m+n+1)}. \tag*{(45)} M G := I m 0 0 2 A G T I n 0 0 2 1 n 1 ∈ Z ( m + n + 1 ) × ( m + n + 1 ) . (45) then M G ∈ P M_G\in\mathcal{P} M G ∈ P κ ^ ( M G ) = maxcut ( G ) − 1 / 4 \widehat{\kappa}(M_G)=\operatorname{maxcut}(G)-1/4 κ ( M G ) = maxcut ( G ) − 1/4
Proof. M G ∈ P M_G \in\mathcal{P} M G ∈ P [82 Theorem 2.3] , κ ^ ( M G ) = max z ∈ R m + n + 1 κ ^ ( M G , z ) \hat{\kappa}(M_G)=\max_{z\in\mathbb{R}^{m+n+1}}\hat{\kappa}(M_G,z) κ ^ ( M G ) = max z ∈ R m + n + 1 κ ^ ( M G , z ) y : = 1 2 ( x + M G x ) ∈ R m + n + 1 y := \frac{1}{2}(x+M_Gx)\in\mathbb{R}^{m+n+1} y := 2 1 ( x + M G x ) ∈ R m + n + 1
x = 2 ( I m + n + 1 + M G ) − 1 y = ( I m − A G T 0 0 I n − 1 n 0 0 1 ) ( y 1 y 2 y 3 ) = ( y 1 − A G T y 2 y 2 − y 3 1 n y 3 ) , y 1 ∈ R m , y 2 ∈ R n , y 3 ∈ R . x=2(I_{m+n+1}+M_G)^{-1}y=
\begin{pmatrix}
I_m & -A_G^{\mathsf{T}} & 0\\
0 & I_n & -\mathbf{1}_n\\
0 & 0 & 1
\end{pmatrix}
\begin{pmatrix}
y_1\\
y_2\\
y_3
\end{pmatrix}
=
\begin{pmatrix}
y_1-A_G^{\mathsf{T}}y_2\\
y_2-y_3\mathbf{1}_n\\
y_3
\end{pmatrix},
\quad y_1\in\mathbb{R}^m,\ y_2\in\mathbb{R}^n,\ y_3\in\mathbb{R}. x = 2 ( I m + n + 1 + M G ) − 1 y = I m 0 0 − A G T I n 0 0 − 1 n 1 y 1 y 2 y 3 = y 1 − A G T y 2 y 2 − y 3 1 n y 3 , y 1 ∈ R m , y 2 ∈ R n , y 3 ∈ R . and
x ∘ M G x = ( y 1 − A G T y 2 y 2 − y 3 1 n y 3 ) ∘ ( y 1 + A G T y 2 y 2 + y 3 1 n y 3 ) = ( y 1 2 − ( A G T y 2 ) 2 y 2 2 − y 3 2 1 n y 3 2 ) . (46) x\circ M_Gx=
\begin{pmatrix}
y_1-A_G^{\mathsf{T}}y_2\\
y_2-y_3\mathbf{1}_n\\
y_3
\end{pmatrix}
\circ
\begin{pmatrix}
y_1+A_G^{\mathsf{T}}y_2\\
y_2+y_3\mathbf{1}_n\\
y_3
\end{pmatrix}
=
\begin{pmatrix}
y_1^2-(A_G^{\mathsf{T}}y_2)^2\\
y_2^2-y_3^2\mathbf{1}_n\\
y_3^2
\end{pmatrix}. \tag*{(46)} x ∘ M G x = y 1 − A G T y 2 y 2 − y 3 1 n y 3 ∘ y 1 + A G T y 2 y 2 + y 3 1 n y 3 = y 1 2 − ( A G T y 2 ) 2 y 2 2 − y 3 2 1 n y 3 2 . (46) Since A G A_G A G G G G A G T 1 n = 0 A_G^{\mathsf{T}}\mathbf{1}_n=0 A G T 1 n = 0 N = ( M G − I m + n + 1 ) / 2 N=(M_G-I_{m+n+1})/2 N = ( M G − I m + n + 1 ) /2
N 2 = ( 0 A G T 0 0 0 1 n 0 0 0 ) 2 = ( 0 0 A G T 1 n 0 0 0 0 0 0 ) = 0 , I m + n + 1 − N T N = ( I m 0 0 0 I n − L G 0 0 0 1 − n ) . N^2=
\begin{pmatrix}
0 & A_G^{\mathsf{T}} & 0\\
0 & 0 & \mathbf{1}_n\\
0 & 0 & 0
\end{pmatrix}^2
=
\begin{pmatrix}
0 & 0 & A_G^{\mathsf{T}}\mathbf{1}_n\\
0 & 0 & 0\\
0 & 0 & 0
\end{pmatrix}
=0,\qquad
I_{m+n+1}-N^{\mathsf{T}}N=
\begin{pmatrix}
I_m & 0 & 0\\
0 & I_n-L_G & 0\\
0 & 0 & 1-n
\end{pmatrix}. N 2 = 0 0 0 A G T 0 0 0 1 n 0 2 = 0 0 0 0 0 0 A G T 1 n 0 0 = 0 , I m + n + 1 − N T N = I m 0 0 0 I n − L G 0 0 0 1 − n . Using Proposition 7.11, we have
x T M G x = y T ( I m + n + 1 − N T N ) y = ∥ y ∥ 2 − y 2 T L G y 2 − n y 3 2 = ∥ y 1 ∥ 2 + y 2 T ( I n − L G ) y 2 − ( n − 1 ) y 3 2 . (47) \begin{aligned}
x^{\mathsf{T}}M_Gx&=y^{\mathsf{T}}(I_{m+n+1}-N^{\mathsf{T}}N)y\\
&=\lVert y\rVert^2-y_2^{\mathsf{T}}L_Gy_2-ny_3^2\\
&=\lVert y_1\rVert^2+y_2^{\mathsf{T}}(I_n-L_G)y_2-(n-1)y_3^2. \tag*{(47)}
\end{aligned} x T M G x = y T ( I m + n + 1 − N T N ) y = ∥ y ∥ 2 − y 2 T L G y 2 − n y 3 2 = ∥ y 1 ∥ 2 + y 2 T ( I n − L G ) y 2 − ( n − 1 ) y 3 2 . (47) Claim 7.14. κ ^ ( M G ) ≥ maxcut ( G ) − 1 / 4 \hat{\kappa}(M_G)\geq\operatorname{maxcut}(G)-1/4 κ ^ ( M G ) ≥ maxcut ( G ) − 1/4
Proof. Let y 2 ∈ { − 1 , 1 } n y_2\in\{-1,1\}^n y 2 ∈ { − 1 , 1 } n maxcut ( G ) \operatorname{maxcut}(G) maxcut ( G ) y 2 T L G y 2 = 4 maxcut ( G ) y_2^{\mathsf{T}}L_Gy_2=4\operatorname{maxcut}(G) y 2 T L G y 2 = 4 maxcut ( G ) y 1 = 0 m y_1=0_m y 1 = 0 m y 3 = 1 y_3=1 y 3 = 1 (46) , we have
x ∘ M G x = ( − ( A G T y 2 ) 2 0 n 1 ) . x\circ M_Gx=
\begin{pmatrix}
-(A_G^{\mathsf{T}}y_2)^2\\
0_n\\
1
\end{pmatrix}. x ∘ M G x = − ( A G T y 2 ) 2 0 n 1 . giving x T M G x = − y 2 T L G y 2 + 1 = − 4 maxcut ( G ) + 1 x^{\mathsf{T}}M_Gx=-y_2^{\mathsf{T}}L_Gy_2+1=-4\operatorname{maxcut}(G)+1 x T M G x = − y 2 T L G y 2 + 1 = − 4 maxcut ( G ) + 1 ∑ i ∈ I M G + ( x ) x i ( M G x ) i = 1 \sum_{i\in I^+_{M_G}(x)}x_i(M_Gx)_i=1 ∑ i ∈ I M G + ( x ) x i ( M G x ) i = 1
x T M G x = − ( 4 maxcut ( G ) − 1 ) ∑ i ∈ I M G + ( x ) x i ( M G x ) i , x^{\mathsf{T}}M_Gx=-(4\operatorname{maxcut}(G)-1)\sum_{i\in I^+_{M_G}(x)}x_i(M_Gx)_i, x T M G x = − ( 4 maxcut ( G ) − 1 ) i ∈ I M G + ( x ) ∑ x i ( M G x ) i , so for x = ( − A G T y 2 , y 2 − 1 n , 1 ) ∈ R m + n + 1 x=(-A_G^{\mathsf{T}}y_2,y_2-\mathbf{1}_n,1)\in\mathbb{R}^{m+n+1} x = ( − A G T y 2 , y 2 − 1 n , 1 ) ∈ R m + n + 1 κ ^ ( M G , x ) = maxcut ( G ) − 1 / 4 \hat{\kappa}(M_G,x)=\operatorname{maxcut}(G)-1/4 κ ^ ( M G , x ) = maxcut ( G ) − 1/4 κ ^ ( M G ) ≥ maxcut ( G ) − 1 / 4 \hat{\kappa}(M_G)\geq\operatorname{maxcut}(G)-1/4 κ ^ ( M G ) ≥ maxcut ( G ) − 1/4 □ \square □
Claim 7.15. κ ^ ( M G ) ≤ maxcut ( G ) − 1 / 4 \hat{\kappa}(M_G)\leq\operatorname{maxcut}(G)-1/4 κ ^ ( M G ) ≤ maxcut ( G ) − 1/4
Proof. From (47) and Proposition 7.12, we get
x T M G x ≥ − ( 4 maxcut ( G ) − n ) ∥ y 2 ∥ ∞ 2 − ( n − 1 ) y 3 2 . (48) x^{\mathsf{T}}M_Gx\geq-(4\operatorname{maxcut}(G)-n)\lVert y_2\rVert_\infty^2-(n-1)y_3^2. \tag*{(48)} x T M G x ≥ − ( 4 maxcut ( G ) − n ) ∥ y 2 ∥ ∞ 2 − ( n − 1 ) y 3 2 . (48) We now consider the positive contribution in x T M G x x^{\mathsf{T}}M_Gx x T M G x x ∘ M G x x\circ M_Gx x ∘ M G x (46) ,
∑ i ∈ I M G + ( x ) x i ( M G x ) i ≥ y 3 2 + ∑ i = 1 n max { y 2 i 2 − y 3 2 , 0 } ≥ max { ∥ y 2 ∥ ∞ 2 , y 3 2 } . (49) \sum_{i\in I^+_{M_G}(x)}x_i(M_Gx)_i\geq y_3^2+\sum_{i=1}^{n}\max\{y_{2i}^2-y_3^2,0\}\geq\max\{\lVert y_2\rVert_\infty^2,y_3^2\}. \tag*{(49)} i ∈ I M G + ( x ) ∑ x i ( M G x ) i ≥ y 3 2 + i = 1 ∑ n max { y 2 i 2 − y 3 2 , 0 } ≥ max {∥ y 2 ∥ ∞ 2 , y 3 2 } . (49) Since maxcut ( G ) ≥ m / 2 \operatorname{maxcut}(G)\geq m/2 maxcut ( G ) ≥ m /2 [26 ] ) and G G G maxcut ( G ) ≥ m / 2 ≥ ( n − 1 ) / 2 ≥ n / 4 \operatorname{maxcut}(G)\geq m/2\geq(n-1)/2\geq n/4 maxcut ( G ) ≥ m /2 ≥ ( n − 1 ) /2 ≥ n /4 (48) and (49) to obtain
x T M G x ≥ − ( 4 maxcut ( G ) − 1 ) ∑ i ∈ I M G + ( x ) x i ( M G x ) i . (50) x^{\mathsf{T}}M_Gx\geq-(4\operatorname{maxcut}(G)-1)\sum_{i\in I^+_{M_G}(x)}x_i(M_Gx)_i. \tag*{(50)} x T M G x ≥ − ( 4 maxcut ( G ) − 1 ) i ∈ I M G + ( x ) ∑ x i ( M G x ) i . (50) This shows that κ ^ ( M G , x ) ≤ maxcut ( G ) − 1 / 4 \hat{\kappa}(M_G,x)\leq\operatorname{maxcut}(G)-1/4 κ ^ ( M G , x ) ≤ maxcut ( G ) − 1/4 x ∈ R m + n + 1 x\in\mathbb{R}^{m+n+1} x ∈ R m + n + 1 κ ^ ( M G ) ≤ maxcut ( G ) − 1 / 4 \hat{\kappa}(M_G)\leq\operatorname{maxcut}(G)-1/4 κ ^ ( M G ) ≤ maxcut ( G ) − 1/4 □ \square □
These claims together show κ ^ ( M G ) = maxcut ( G ) − 1 / 4 \widehat{\kappa}(M_G)=\operatorname{maxcut}(G)-1/4 κ ( M G ) = maxcut ( G ) − 1/4
Proof of Theorem 1.4 . As in Lemma 7.13, we can construct M G ∈ Z ( m + n + 1 ) × ( m + n + 1 ) M_G \in\mathbb{Z}^{(m+n+1)\times(m+n+1)} M G ∈ Z ( m + n + 1 ) × ( m + n + 1 ) G G G ∣ V ∣ = n ≥ 2 |V|=n\ge2 ∣ V ∣ = n ≥ 2 ∣ E ∣ = m |E|=m ∣ E ∣ = m M G ∈ P M_G\in\mathcal{P} M G ∈ P κ ^ ( M G ) = maxcut ( G ) − 1 / 4 \widehat{\kappa}(M_G)=\operatorname{maxcut}(G)-1/4 κ ( M G ) = maxcut ( G ) − 1/4 1 / 4 + κ ^ ( M ) 1/4+\widehat{\kappa}(M) 1/4 + κ ( M ) M M M α \alpha α G G G
Rump’s characterization of sufficient matrix
Having seen Rump’s original characterization of P \mathcal{P} P P 0 \mathcal{P}_0 P 0 P ⋆ \mathcal{P}_\star P ⋆
Throughout this section, we assume M ∈ P 0 M\in\mathcal{P}_0 M ∈ P 0 I n + M I_n+M I n + M Corollary 2.5 ). We again make use of the change of variable by midpoint in Proposition 7.11 to express v ∘ ( M v ) v\circ(Mv) v ∘ ( M v ) x : = 1 2 ( v + M v ) x:=\frac{1}{2}(v+Mv) x := 2 1 ( v + M v ) N : = 1 2 ( M − I n ) N:=\frac{1}{2}(M-I_n) N := 2 1 ( M − I n ) Proposition 7.11 ,
M v = ( I n + N ( I n + N ) − 1 ) x , v = ( I n − N ( I n + N ) − 1 ) x , (51) Mv=(I_n+N(I_n+N)^{-1})x,\qquad v=(I_n-N(I_n+N)^{-1})x, \tag*{(51)} M v = ( I n + N ( I n + N ) − 1 ) x , v = ( I n − N ( I n + N ) − 1 ) x , (51) where N ( I n + N ) − 1 = ( M − I n ) ( M + I n ) − 1 = C − 1 ( M ) N(I_n+N)^{-1}=(M-I_n)(M+I_n)^{-1}=C^{-1}(M) N ( I n + N ) − 1 = ( M − I n ) ( M + I n ) − 1 = C − 1 ( M ) M M M Proposition 7.3 ). Hence, we have
v ∘ ( M v ) = ( I − C − 1 ( M ) ) x ∘ ( I + C − 1 ( M ) ) x = x 2 − ( C − 1 ( M ) x ) 2 . (52) v\circ(Mv)=(I-C^{-1}(M))x\circ(I+C^{-1}(M))x=x^2-(C^{-1}(M)x)^2. \tag*{(52)} v ∘ ( M v ) = ( I − C − 1 ( M )) x ∘ ( I + C − 1 ( M )) x = x 2 − ( C − 1 ( M ) x ) 2 . (52) We recall M M M v ∘ ( M v ) ≤ 0 n v\circ(Mv)\leq0_n v ∘ ( M v ) ≤ 0 n v ∘ ( M v ) = 0 n v\circ(Mv)=0_n v ∘ ( M v ) = 0 n v ∈ R n v\in\mathbb{R}^n v ∈ R n M T M^{\mathsf{T}} M T M ∈ P ⋆ M\in\mathcal{P}_\star M ∈ P ⋆ M M M [19 Theorem 1.4] also gives a strengthening of Väliaho’s theorem, characterizing sufficient matrices using both v ∘ M v v\circ Mv v ∘ M v u ∘ M v + v ∘ M u u\circ Mv+v\circ Mu u ∘ M v + v ∘ M u
Theorem 7.16 ([19 ] ). For any M ∈ R n × n M\in\mathbb{R}^{n\times n} M ∈ R n × n M M M u , v ∈ R n u,v\in\mathbb{R}^n u , v ∈ R n v ∘ M v ≤ 0 n v\circ Mv\leq0_n v ∘ M v ≤ 0 n u ∘ M v + v ∘ M u ≤ 0 n u\circ Mv+v\circ Mu\leq0_n u ∘ M v + v ∘ M u ≤ 0 n v ∘ M v = 0 n v\circ Mv=0_n v ∘ M v = 0 n u ∘ M v + v ∘ M u = 0 n u\circ Mv+v\circ Mu=0_n u ∘ M v + v ∘ M u = 0 n
Similar to (52) , we can express u ∘ M v + v ∘ M u u\circ Mv+v\circ Mu u ∘ M v + v ∘ M u x = 1 2 ( v + M v ) x=\frac{1}{2}(v+Mv) x = 2 1 ( v + M v ) N = 1 2 ( M − I n ) N=\frac{1}{2}(M-I_n) N = 2 1 ( M − I n ) y = 1 2 ( u + M u ) y=\frac{1}{2}(u+Mu) y = 2 1 ( u + M u ) (51) and C − 1 ( M ) = N ( I n + N ) − 1 C^{-1}(M)=N(I_n+N)^{-1} C − 1 ( M ) = N ( I n + N ) − 1
u ∘ M v + v ∘ M u = ( I n − C − 1 ( M ) ) x ∘ ( I n + C − 1 ( M ) ) x + ( I n − C − 1 ( M ) ) x ∘ ( I n + C − 1 ( M ) ) y = 2 ( x ∘ y − C − 1 ( M ) x ∘ C − 1 ( M ) y ) . (53) \begin{aligned}
u\circ Mv+v\circ Mu={}&(I_n-C^{-1}(M))x\circ(I_n+C^{-1}(M))x+(I_n-C^{-1}(M))x\circ(I_n+C^{-1}(M))y\\
={}&2(x\circ y-C^{-1}(M)x\circ C^{-1}(M)y). \tag*{(53)}
\end{aligned} u ∘ M v + v ∘ M u = = ( I n − C − 1 ( M )) x ∘ ( I n + C − 1 ( M )) x + ( I n − C − 1 ( M )) x ∘ ( I n + C − 1 ( M )) y 2 ( x ∘ y − C − 1 ( M ) x ∘ C − 1 ( M ) y ) . (53) We can now give a full characterization of P \mathcal{P} P P ⋆ \mathcal{P}_\star P ⋆ P 0 \mathcal{P}_0 P 0
Theorem 7.17 (Rump’s theorem for P ⋆ \mathcal{P}_\star P ⋆ M ∈ R n × n M\in\mathbb{R}^{n\times n} M ∈ R n × n I n + M I_n+M I n + M M ∉ P 0 M\notin\mathcal{P}_0 M ∈ / P 0 I n + M I_n+M I n + M M ~ : = C − 1 ( M ) = ( M − I n ) ( M + I n ) − 1 \widetilde{M}:=C^{-1}(M)=(M-I_n)(M+I_n)^{-1} M := C − 1 ( M ) = ( M − I n ) ( M + I n ) − 1
(i) M ∈ P M\in\mathcal{P} M ∈ P ρ 0 S ( M ~ ) < 1 \rho_0^{\mathsf{S}}(\widetilde{M})<1 ρ 0 S ( M ) < 1
(ii) M ∈ P ⋆ ∖ P M\in\mathcal{P}_\star\setminus\mathcal{P} M ∈ P ⋆ ∖ P ρ 0 S ( M ~ ) = 1 \rho_0^{\mathsf{S}}(\widetilde{M})=1 ρ 0 S ( M ) = 1 x ∈ R n x\in\mathbb{R}^n x ∈ R n ∣ M ~ x ∣ ≥ ∣ x ∣ |\widetilde{M}x|\geq|x| ∣ M x ∣ ≥ ∣ x ∣ ∣ M ~ x ∣ = ∣ x ∣ |\widetilde{M}x|=|x| ∣ M x ∣ = ∣ x ∣ x , y ∈ R n x,y\in\mathbb{R}^n x , y ∈ R n ∣ M ~ x ∣ = ∣ x ∣ |\widetilde{M}x|=|x| ∣ M x ∣ = ∣ x ∣ M ~ x ∘ M ~ y ≥ x ∘ y \widetilde{M}x\circ\widetilde{M}y\geq x\circ y M x ∘ M y ≥ x ∘ y M ~ x ∘ M ~ y = x ∘ y \widetilde{M}x\circ\widetilde{M}y=x\circ y M x ∘ M y = x ∘ y
(iii) M ∈ P 0 M\in\mathcal{P}_0 M ∈ P 0 ρ 0 S ( M ~ ) ≤ 1 \rho_0^{\mathsf{S}}(\widetilde{M})\leq1 ρ 0 S ( M ) ≤ 1
Proof. By Corollary 2.5 , M ∈ P 0 M\in\mathcal{P}_0 M ∈ P 0 I n + M I_n+M I n + M Theorem 7.4 and Theorem 7.6. Using Theorem 7.16 , (52) , and (53) , M ∈ P ⋆ ∖ P M\in\mathcal{P}_\star\setminus\mathcal{P} M ∈ P ⋆ ∖ P ρ 0 S ( M ~ ) = 1 \rho_0^{\mathsf{S}}(\widetilde{M})=1 ρ 0 S ( M ) = 1 x , y ∈ R n x,y\in\mathbb{R}^n x , y ∈ R n ∣ M ~ x ∣ ≥ ∣ x ∣ |\widetilde{M}x|\geq|x| ∣ M x ∣ ≥ ∣ x ∣ M ~ x ∘ M ~ y ≥ x ∘ y \widetilde{M}x\circ\widetilde{M}y\geq x\circ y M x ∘ M y ≥ x ∘ y ∣ M ~ x ∣ = ∣ x ∣ |\widetilde{M}x|=|x| ∣ M x ∣ = ∣ x ∣ M ~ x ∘ M ~ y = x ∘ y \widetilde{M}x\circ\widetilde{M}y=x\circ y M x ∘ M y = x ∘ y
Criteria for a general matrix to be sufficient were extensively studied by Väliaho [80 ] . Apart from providing a spectral viewpoint of P ⋆ \mathcal{P}_\star P ⋆ Theorem 7.17 may also be used to develop new properties of sufficient matrices, which then give practical screening tests to detect non-sufficiency.
For example, Theorem 7.17 yields the following corollary on the structure of the zero eigenspace of a sufficient matrix. An eigenvalue is semi-simple if its algebraic multiplicity (possibly zero) is equal to its geometric multiplicity, that is, the associated Jordan block of that eigenvalue is diagonal. For x ∈ R n x\in\mathbb{R}^n x ∈ R n sgn ( x ) ∈ { + 1 , 0 } n \operatorname{sgn}(x)\in\{+1,0\}^n sgn ( x ) ∈ { + 1 , 0 } n ( sgn ( x ) ) i = sgn ( x i ) (\operatorname{sgn}(x))_i=\operatorname{sgn}(x_i) ( sgn ( x ) ) i = sgn ( x i ) i ∈ [ n ] i\in[n] i ∈ [ n ] sgn ( 0 ) : = 0 \operatorname{sgn}(0):=0 sgn ( 0 ) := 0 U ⊆ R n U\subseteq\mathbb{R}^n U ⊆ R n sgn ( U ) : = { sgn ( x ) : x ∈ U } \operatorname{sgn}(U):=\{\operatorname{sgn}(x):x\in U\} sgn ( U ) := { sgn ( x ) : x ∈ U }
Corollary 7.18. Let M ∈ P ⋆ ∩ R n × n M\in\mathcal{P}_\star\cap\mathbb{R}^{n\times n} M ∈ P ⋆ ∩ R n × n 0 0 0 semi-simple eigenvalue of M M M ker ( M p ) = ker ( M ) \ker(M^p)=\ker(M) ker ( M p ) = ker ( M ) rk ( M p ) = rk ( M ) \operatorname{rk}(M^p)=\operatorname{rk}(M) rk ( M p ) = rk ( M ) p ∈ N p\in\mathbb{N} p ∈ N sgn ( ker ( M ) ) = sgn ( ker ( M T ) ) \operatorname{sgn}(\ker(M))=\operatorname{sgn}(\ker(M^{\mathsf{T}})) sgn ( ker ( M )) = sgn ( ker ( M T ))
The proof can be found in § 9.3 .
Initialization, termination and guessing of the handicap number
We start by two simple statements on (column) sufficient matrices that will be needed for the initialization scheme. The first one is true for any sufficient matrix, and follows easily by the convexity of the solution set of LCP(M , q M,q M , q [8 Theorem 3.5.8] .
Lemma 8.1. Let M ∈ P ⋆ ∩ R n × n M \in\mathcal{P}_{\star} \cap\mathbb{R}^{n \times n} M ∈ P ⋆ ∩ R n × n q ∈ R n q \in\mathbb{R}^{n} q ∈ R n ( x , s ) (x,s) ( x , s ) ( x ′ , s ′ ) (x',s') ( x ′ , s ′ ) M , q M,q M , q i ∈ [ n ] i \in[n] i ∈ [ n ] x i = x i ′ x_i=x'_i x i = x i ′ s i = s i ′ s_i=s'_i s i = s i ′
For sufficient matrices, the above statement is also implied by the next bound, that can be seen as a generalization of the near monotonicity of the central path of linear programs, see e.g., [84 Lemma 16] . We will use this to identify the optimal support from a sufficiently good approximate solution.
Lemma 8.2. Let M ∈ P ⋆ ( κ ) M \in\mathcal{P}_{\star}(\kappa) M ∈ P ⋆ ( κ ) q ∈ R n q \in\mathbb{R}^{n} q ∈ R n ( x , s , μ ) ∈ N 2 ( 1 / 2 ) (x,s,\mu) \in\mathcal{N}^{2}(1/2) ( x , s , μ ) ∈ N 2 ( 1/2 ) ( x ˉ , s ˉ ) (\bar{x},\bar{s}) ( x ˉ , s ˉ ) M , q M,q M , q
( x ˉ , s ˉ ) ≤ 32 ( κ + 1 ) n ⋅ ( x , s ) . (\bar{x},\bar{s}) \le32(\kappa+1)n \cdot(x,s). ( x ˉ , s ˉ ) ≤ 32 ( κ + 1 ) n ⋅ ( x , s ) . Proof. Let v : = x − x ˉ v:=x-\bar{x} v := x − x ˉ
v ( M v ) = ( x − x ˉ ) ( s − s ˉ ) = x s + x ˉ s ˉ − x ˉ s − x s ˉ ≤ 3 2 μ ⋅ 1 n − x ˉ s − s ˉ x ≤ 3 2 μ ⋅ 1 n , v(Mv)=(x-\bar{x})(s-\bar{s})=xs+\bar{x}\bar{s}-\bar{x}s-x\bar{s}\le\frac{3}{2}\mu\cdot\mathbf{1}_{n}-\bar{x}s-\bar{s}x\le\frac{3}{2}\mu\cdot\mathbf{1}_{n}, v ( M v ) = ( x − x ˉ ) ( s − s ˉ ) = x s + x ˉ s ˉ − x ˉ s − x s ˉ ≤ 2 3 μ ⋅ 1 n − x ˉ s − s ˉ x ≤ 2 3 μ ⋅ 1 n , using the bound on x s ≤ 3 2 μ ⋅ 1 n xs \le\frac{3}{2}\mu\cdot\mathbf{1}_{n} x s ≤ 2 3 μ ⋅ 1 n ( x , s , μ ) ∈ N 2 ( 1 / 2 ) (x,s,\mu)\in\mathcal{N}^{2}(1/2) ( x , s , μ ) ∈ N 2 ( 1/2 ) x ˉ s ˉ = 0 \bar{x}\bar{s}=0 x ˉ s ˉ = 0 κ \kappa κ
0 ≤ ( 4 κ + 1 ) ∑ i ∈ I M ( v ) + v i ( M v ) i + ∑ i ∈ I M ( v ) − v i ( M v ) i ≤ 3 2 ( 4 κ + 1 ) ∣ I M ( v ) + ∣ μ + ∑ i ∈ I M ( v ) − v i ( M v ) i , 0\le(4\kappa+1)\sum_{i\in I^{+}_{M(v)}}v_i(Mv)_i+\sum_{i\in I^{-}_{M(v)}}v_i(Mv)_i\le\frac{3}{2}(4\kappa+1)|I^{+}_{M(v)}|\mu+\sum_{i\in I^{-}_{M(v)}}v_i(Mv)_i, 0 ≤ ( 4 κ + 1 ) i ∈ I M ( v ) + ∑ v i ( M v ) i + i ∈ I M ( v ) − ∑ v i ( M v ) i ≤ 2 3 ( 4 κ + 1 ) ∣ I M ( v ) + ∣ μ + i ∈ I M ( v ) − ∑ v i ( M v ) i , implying that for any i ∈ [ n ] i\in[n] i ∈ [ n ]
− 3 2 ( 4 κ + 1 ) ( n − 1 ) μ ≤ v i ( M v ) i ≤ 3 2 μ ⋅ 1 n − x ˉ i s i − s ˉ i x i , -\frac{3}{2}(4\kappa+1)(n-1)\mu\le v_i(Mv)_i\le\frac{3}{2}\mu\cdot\mathbf{1}_{n}-\bar{x}_i s_i-\bar{s}_i x_i, − 2 3 ( 4 κ + 1 ) ( n − 1 ) μ ≤ v i ( M v ) i ≤ 2 3 μ ⋅ 1 n − x ˉ i s i − s ˉ i x i , giving
x ˉ i s i + s ˉ i x i ≤ 3 ( 4 κ + 1 ) n μ . \bar{x}_i s_i+\bar{s}_i x_i\le3(4\kappa+1)n\mu. x ˉ i s i + s ˉ i x i ≤ 3 ( 4 κ + 1 ) n μ . By ( x , s , μ ) ∈ N 2 ( 1 / 2 ) (x,s,\mu)\in\mathcal{N}^{2}(1/2) ( x , s , μ ) ∈ N 2 ( 1/2 ) x i s i ≥ μ / 2 x_i s_i\ge\mu/2 x i s i ≥ μ /2
x ˉ i x i + s ˉ i s i ≤ 6 ( 4 κ + 1 ) n , \frac{\bar{x}_i}{x_i}+\frac{\bar{s}_i}{s_i}\le6(4\kappa+1)n, x i x ˉ i + s i s ˉ i ≤ 6 ( 4 κ + 1 ) n , implying the claim.
The extended system
To find an initial point ( x 0 , s 0 ) (x^{0},s^{0}) ( x 0 , s 0 ) § 5 of [38 ] , by considering an LCP(M ^ , q ^ \widehat{M},\widehat{q} M , q 2 n 2n 2 n M , q M,q M , q M , q M,q M , q [70 ] .
Lemma 8.3. M ∈ Q n × n M\in\mathbb{Q}^{n\times n} M ∈ Q n × n q ∈ Q n q\in\mathbb{Q}^{n} q ∈ Q n L L L ( M , q ) (M,q) ( M , q ) Δ ≤ 2 O ( L ) \Delta\le2^{O(L)} Δ ≤ 2 O ( L ) Δ ∈ Z \Delta\in\mathbb{Z} Δ ∈ Z ∥ M ∥ max + ∥ q ∥ ∞ ≤ Δ \|M\|_{\max}+\|q\|_{\infty}\le\Delta ∥ M ∥ m a x + ∥ q ∥ ∞ ≤ Δ ( x , s ) (x,s) ( x , s ) F ( M , q ) \mathcal{F}(M,q) F ( M , q ) i ∈ [ n ] i\in[n] i ∈ [ n ] x i , s i ∈ { 0 } ∪ [ 1 / Δ , Δ ] x_i,s_i\in\{0\}\cup[1/\Delta,\Delta] x i , s i ∈ { 0 } ∪ [ 1/Δ , Δ ]
We say that ( x , s ) (x,s) ( x , s ) M , q M,q M , q F ( M , q ) \mathcal{F}(M,q) F ( M , q ) ( x , s ) (x,s) ( x , s ) M , q M,q M , q ( x ˉ , s ˉ ) (\bar{x},\bar{s}) ( x ˉ , s ˉ ) supp ( x ˉ ) ⊆ supp ( x ) \operatorname{supp}(\bar{x})\subseteq\operatorname{supp}(x) supp ( x ˉ ) ⊆ supp ( x ) supp ( s ˉ ) ⊆ supp ( s ) \operatorname{supp}(\bar{s})\subseteq\operatorname{supp}(s) supp ( s ˉ ) ⊆ supp ( s )
The embedding is given as
M ^ x ^ + q ^ = s ^ , x ^ T s ^ = 0 , x ^ , s ^ ≥ 0 2 n , (LCP ( M ^ , q ^ ) ) \widehat{M}\widehat{x}+\widehat{q}=\widehat{s},\qquad\widehat{x}^{\mathsf{T}}\widehat{s}=0,\qquad\widehat{x},\widehat{s}\ge0_{2n},\qquad\text{(LCP}(\widehat{M},\widehat{q})\text{)} M x + q = s , x T s = 0 , x , s ≥ 0 2 n , (LCP ( M , q ) ) where
M ^ = ( M I n − I n 0 ) , q ^ = ( q q ~ ) , x ^ = ( x x ~ ) , s ^ = ( s s ~ ) . \widehat{M}=\begin{pmatrix} M & I_n \\ -I_n & 0 \end{pmatrix},\qquad\widehat{q}=\begin{pmatrix} q \\ \widetilde{q} \end{pmatrix},\qquad\widehat{x}=\begin{pmatrix} x \\ \widetilde{x} \end{pmatrix},\qquad\widehat{s}=\begin{pmatrix} s \\ \widetilde{s} \end{pmatrix}. M = ( M − I n I n 0 ) , q = ( q q ) , x = ( x x ) , s = ( s s ) . Lemma 8.4 (Initialization). Let M ∈ P ⋆ ∩ Q n × n M \in\mathcal{P}_{\star} \cap\mathbb{Q}^{n \times n} M ∈ P ⋆ ∩ Q n × n q ∈ Q n q \in\mathbb{Q}^{n} q ∈ Q n L L L ( M , q ) (M,q) ( M , q ) Δ ≤ 2 O ( L ) \Delta\le2^{O(L)} Δ ≤ 2 O ( L ) x ^ 0 = ( x 0 , x ~ 0 ) \hat{x}^{0}=(x^{0},\tilde{x}^{0}) x ^ 0 = ( x 0 , x ~ 0 ) s ^ 0 = ( s 0 , s ~ 0 ) \hat{s}^{0}=(s^{0},\tilde{s}^{0}) s ^ 0 = ( s 0 , s ~ 0 ) q ~ \tilde{q} q ~ μ ^ 0 \hat{\mu}_{0} μ ^ 0
x 0 = 2 Δ 1 n , s 0 = 4 Δ 2 n 1 n , x ~ 0 = s 0 − M x 0 − q , s ~ 0 = 8 n Δ 3 x ~ 0 , q ~ = x 0 + s ~ 0 , μ ^ 0 = 8 n Δ 3 . (54) x^{0}=2\Delta\mathbf{1}_{n}, \qquad s^{0}=4\Delta^{2}n\mathbf{1}_{n}, \qquad\tilde{x}^{0}=s^{0}-Mx^{0}-q, \qquad\tilde{s}^{0}=\frac{8n\Delta^{3}}{\tilde{x}^{0}}, \qquad\tilde{q}=x^{0}+\tilde{s}^{0}, \qquad\hat{\mu}_{0}=8n\Delta^{3}. \tag*{(54)} x 0 = 2Δ 1 n , s 0 = 4 Δ 2 n 1 n , x ~ 0 = s 0 − M x 0 − q , s ~ 0 = x ~ 0 8 n Δ 3 , q ~ = x 0 + s ~ 0 , μ ^ 0 = 8 n Δ 3 . (54) M ∈ P ⋆ ( κ ) M \in\mathcal{P}_{\star}(\kappa) M ∈ P ⋆ ( κ ) M ^ ∈ P ⋆ ( κ ) \hat{M} \in\mathcal{P}_{\star}(\kappa) M ^ ∈ P ⋆ ( κ ) κ ^ ( M ) = κ ^ ( M ^ ) \hat{\kappa}(M)=\hat{\kappa}(\hat{M}) κ ^ ( M ) = κ ^ ( M ^ ) ( x ^ 0 , s ^ 0 ) (\hat{x}^{0},\hat{s}^{0}) ( x ^ 0 , s ^ 0 ) μ ^ 0 \hat{\mu}_{0} μ ^ 0
Let ( x ^ ∗ , s ^ ∗ ) = ( ( x ∗ , x ~ ∗ ) , ( s ∗ , s ~ ∗ ) ) (\hat{x}^{*},\hat{s}^{*})=((x^{*},\tilde{x}^{*}),(s^{*},\tilde{s}^{*})) ( x ^ ∗ , s ^ ∗ ) = (( x ∗ , x ~ ∗ ) , ( s ∗ , s ~ ∗ )) x ~ ∗ = 0 n \tilde{x}^{*}=0_{n} x ~ ∗ = 0 n ( x ∗ , s ∗ ) (x^{*},s^{*}) ( x ∗ , s ∗ ) x ~ ∗ ≠ 0 n \tilde{x}^{*}\ne0_{n} x ~ ∗ = 0 n
Proof. The proof of M ∈ P ⋆ ( κ ) M \in\mathcal{P}_{\star}(\kappa) M ∈ P ⋆ ( κ ) M ^ ∈ P ⋆ ( κ ) \hat{M} \in\mathcal{P}_{\star}(\kappa) M ^ ∈ P ⋆ ( κ ) [38 ] , Lemma 5.3, implying κ ^ ( M ) = κ ^ ( M ^ ) \hat{\kappa}(M)=\hat{\kappa}(\hat{M}) κ ^ ( M ) = κ ^ ( M ^ ) s ~ 0 = M ^ x ^ 0 + q ^ \tilde{s}^{0}=\hat{M}\hat{x}^{0}+\hat{q} s ~ 0 = M ^ x ^ 0 + q ^ x ^ 0 s ^ 0 = 8 n Δ 3 1 2 n \hat{x}^{0}\hat{s}^{0}=8n\Delta^{3}\mathbf{1}_{2n} x ^ 0 s ^ 0 = 8 n Δ 3 1 2 n
x ~ 0 ≥ s 0 − ∥ M x 0 + q ∥ ∞ 1 n ≥ ( 4 Δ 2 n − 2 Δ 2 n − Δ ) 1 n > 0 n , \tilde{x}^{0}\ge s^{0}-\lVert Mx^{0}+q\rVert_{\infty}\mathbf{1}_{n}\ge(4\Delta^{2}n-2\Delta^{2}n-\Delta)\mathbf{1}_{n}>0_{n}, x ~ 0 ≥ s 0 − ∥ M x 0 + q ∥ ∞ 1 n ≥ ( 4 Δ 2 n − 2 Δ 2 n − Δ ) 1 n > 0 n , Hence, ( x ^ 0 , s ^ 0 ) (\hat{x}^{0},\hat{s}^{0}) ( x ^ 0 , s ^ 0 ) μ ^ 0 = 8 n Δ 3 \hat{\mu}_{0}=8n\Delta^{3} μ ^ 0 = 8 n Δ 3 x ~ ∗ = 0 n \tilde{x}^{*}=0_{n} x ~ ∗ = 0 n ( x ∗ , s ∗ ) (x^{*},s^{*}) ( x ∗ , s ∗ )
If x ~ ∗ ≠ 0 n \tilde{x}^{*}\ne0_{n} x ~ ∗ = 0 n i ∈ [ n ] i \in[n] i ∈ [ n ] x ~ i ∗ > 0 \tilde{x}^{*}_{i}>0 x ~ i ∗ > 0 ( x ˉ , s ˉ ) (\bar{x},\bar{s}) ( x ˉ , s ˉ ) ( ( x ˉ , 0 n ) , ( s ˉ , s ~ ) ) ((\bar{x},0_{n}),(\bar{s},\tilde{s})) (( x ˉ , 0 n ) , ( s ˉ , s ~ )) s ~ : = q ~ − x ˉ \tilde{s}:=\tilde{q}-\bar{x} s ~ := q ~ − x ˉ x ~ i = 0 < x ~ i ∗ \tilde{x}_{i}=0<\tilde{x}^{*}_{i} x ~ i = 0 < x ~ i ∗ s ~ i = s ~ i ∗ = 0 \tilde{s}_{i}=\tilde{s}^{*}_{i}=0 s ~ i = s ~ i ∗ = 0 x ~ i ∗ s ~ i ∗ = 0 \tilde{x}^{*}_{i}\tilde{s}^{*}_{i}=0 x ~ i ∗ s ~ i ∗ = 0
s ~ i = q ~ i − x ˉ i ≥ x i 0 − x ˉ i = 2 Δ − x ˉ i > 0 , (55) \tilde{s}_{i}=\tilde{q}_{i}-\bar{x}_{i}\ge x^{0}_{i}-\bar{x}_{i}=2\Delta-\bar{x}_{i}>0, \tag*{(55)} s ~ i = q ~ i − x ˉ i ≥ x i 0 − x ˉ i = 2Δ − x ˉ i > 0 , (55) by Lemma 8.3, a contradiction.
We now use the near-monotonicity property Lemma 8.2 to identify the optimal support of LCP(M,q) from an approximate solution to the embedded system, assuming that M ∈ P ⋆ ( κ ) M \in\mathcal{P}_{\star}(\kappa) M ∈ P ⋆ ( κ )
Lemma 8.5. Let M ∈ Q n × n M \in\mathbb{Q}^{n \times n} M ∈ Q n × n q ∈ Q n q \in\mathbb{Q}^{n} q ∈ Q n κ > 0 \kappa>0 κ > 0 L L L ( M , q ) (M,q) ( M , q ) Δ ≤ 2 O ( L ) \Delta\le2^{O(L)} Δ ≤ 2 O ( L ) q ~ \tilde{q} q ~ (54) , and let ( ( x , x ~ ) , ( s , s ~ ) , μ ) ∈ N 2 ( 1 / 2 ) ((x,\tilde{x}),(s,\tilde{s}),\mu)\in\mathcal{N}^{2}(1/2) (( x , x ~ ) , ( s , s ~ ) , μ ) ∈ N 2 ( 1/2 ) μ ≤ 1 / ( 16384 n 3 Δ 2 ( κ + 1 ) 2 ) \mu\le1/(16384n^{3}\Delta^{2}(\kappa+1)^{2}) μ ≤ 1/ ( 16384 n 3 Δ 2 ( κ + 1 ) 2 )
I : = { i ∈ [ n ] : x i ≤ 2 n μ } , J : = { j ∈ [ n ] : s j ≤ 2 n μ } . I:=\{i\in[n]:x_{i}\le\sqrt{2n\mu}\},\qquad J:=\{j\in[n]:s_{j}\le\sqrt{2n\mu}\}. I := { i ∈ [ n ] : x i ≤ 2 n μ } , J := { j ∈ [ n ] : s j ≤ 2 n μ } . Then I ∪ J = [ n ] I\cup J=[n] I ∪ J = [ n ]
There exists a solution ( x ˉ , s ˉ ) ∈ F ( M , q ) (\bar{x},\bar{s})\in\mathcal{F}(M,q) ( x ˉ , s ˉ ) ∈ F ( M , q ) x ˉ I = 0 I \bar{x}_{I}=0_{I} x ˉ I = 0 I s ˉ J = 0 J \bar{s}_{J}=0_{J} s ˉ J = 0 J
LCP(M,q) is infeasible;
κ ^ ( M ) > κ \hat{\kappa}(M)>\kappa κ ^ ( M ) > κ
Proof. Since ( ( x , x ~ ) , ( s , s ~ ) , μ ) ∈ N 2 ( 1 / 2 ) ((x,\tilde{x}),(s,\tilde{s}),\mu)\in\mathcal{N}^{2}(1/2) (( x , x ~ ) , ( s , s ~ ) , μ ) ∈ N 2 ( 1/2 ) n n n ∣ x T s / μ − n ∣ ≤ ∥ err ( x , s , μ ) ∥ 1 ≤ n / 2 \lvert x^{\mathsf{T}}s/\mu-n\rvert\le\lVert\operatorname{err}(x,s,\mu)\rVert_{1}\le\sqrt{n}/2 ∣ x T s / μ − n ∣ ≤ ∥ err ( x , s , μ ) ∥ 1 ≤ n /2 x T s ≤ 2 n μ x^{\mathsf{T}}s\le2n\mu x T s ≤ 2 n μ i ∈ [ n ] i\in[n] i ∈ [ n ] 0 ≤ x i s i ≤ 2 n μ 0\le x_{i}s_{i}\le2n\mu 0 ≤ x i s i ≤ 2 n μ I ∪ J = [ n ] I\cup J=[n] I ∪ J = [ n ] κ ^ ( M ) ≤ κ \hat{\kappa}(M)\le\kappa κ ^ ( M ) ≤ κ M ∈ P ⋆ ( κ ) M\in\mathcal{P}_{\star}(\kappa) M ∈ P ⋆ ( κ )
Let ( x ˉ , s ˉ ) (\bar{x},\bar{s}) ( x ˉ , s ˉ ) ( ( x ˉ , 0 n ) , ( s ˉ , q ~ − x ˉ ) ) ((\bar{x},0_{n}),(\bar{s},\tilde{q}-\bar{x})) (( x ˉ , 0 n ) , ( s ˉ , q ~ − x ˉ )) q ~ − x ˉ > 0 n \tilde{q}-\bar{x}>0_{n} q ~ − x ˉ > 0 n ( ( x , x ~ ) , ( s , s ~ ) ) ((x,\tilde{x}),(s,\tilde{s})) (( x , x ~ ) , ( s , s ~ )) ( ( x ˉ , 0 n ) , ( s ˉ , q ~ − x ˉ ) ) ((\bar{x},0_{n}),(\bar{s},\tilde{q}-\bar{x})) (( x ˉ , 0 n ) , ( s ˉ , q ~ − x ˉ )) κ ^ ( M ^ ) = κ ^ ( M ) ≤ κ \hat{\kappa}(\hat{M})=\hat{\kappa}(M)\le\kappa κ ^ ( M ^ ) = κ ^ ( M ) ≤ κ
x ˉ i ≤ 64 ( κ + 1 ) n x i ≤ 2 n μ ⋅ 64 ( κ + 1 ) n < 1 Δ , ∀ i ∈ I , \bar{x}_{i}\le64(\kappa+1)nx_{i}\le\sqrt{2n\mu}\cdot64(\kappa+1)n<\frac{1}{\Delta},\qquad\forall i\in I, x ˉ i ≤ 64 ( κ + 1 ) n x i ≤ 2 n μ ⋅ 64 ( κ + 1 ) n < Δ 1 , ∀ i ∈ I , and
s ˉ i ≤ 64 ( κ + 1 ) n s i ≤ 2 n μ ⋅ 64 ( κ + 1 ) n < 1 Δ , ∀ i ∈ J . \bar{s}_{i}\le64(\kappa+1)ns_{i}\le\sqrt{2n\mu}\cdot64(\kappa+1)n<\frac{1}{\Delta},\qquad\forall i\in J. s ˉ i ≤ 64 ( κ + 1 ) n s i ≤ 2 n μ ⋅ 64 ( κ + 1 ) n < Δ 1 , ∀ i ∈ J . By Lemma 8.3, x ˉ I = 0 I \bar{x}_{I}=0_{I} x ˉ I = 0 I s ˉ J = 0 J \bar{s}_{J}=0_{J} s ˉ J = 0 J
:::
The overall algorithm
Algorithm 3 presents the full version of our algorithm needed to prove Theorem 1.2, where we do not assume knowledge of κ ^ ( M ) \hat{\kappa}(M) κ ^ ( M ) § 7 , given an instance of LCP ( M , q ) \operatorname{LCP}(M,q) LCP ( M , q ) M M M M ∈ P ⋆ M \in\mathcal{P}_{\star} M ∈ P ⋆ [56 ] to handle the unknown handicap number and Illés, E.-Nagy and Terlaky [33 ] to adapt the algorithm to treat general matrices, we can modify our algorithm to require no knowledge of κ ^ ( M ) \hat{\kappa}(M) κ ^ ( M ) M ∈ P ⋆ M \in\mathcal{P}_{\star} M ∈ P ⋆ P ⋆ \mathcal{P}_{\star} P ⋆
Theorem 8.6 ([19 ] ). For any M ∈ P ⋆ ∩ Q n × n M \in\mathcal{P}_{\star} \cap\mathbb{Q}^{n \times n} M ∈ P ⋆ ∩ Q n × n with encoding length length ( M ) \operatorname{length}(M) length ( M ) κ ^ ( M ) < 2 O ( n length ( M ) ) \hat{\kappa}(M) < 2^{O(n\operatorname{length}(M))} κ ^ ( M ) < 2 O ( n length ( M ))
In order to get the better of the running times in Theorem 4.2 and Theorem 6.4, for a guess κ ≥ 1 \kappa\ge1 κ ≥ 1 κ ^ ( M ) \hat{\kappa}(M) κ ^ ( M ) β n ( κ ) \beta_n(\kappa) β n ( κ ) δ n ( κ ) \delta_n(\kappa) δ n ( κ )
β n ( κ ) ∈ { [ 1 8 ( 1 + κ ) , 1 4 ( 1 + κ ) ] , if κ 4 ≤ n 5 , [ 1 8192 n 3 / 2 , 1 4096 n 3 / 2 ] , if κ 4 > n 5 δ n ( κ ) ∈ { [ 1 32 n ( 1 + κ ) , 1 16 n ( 1 + κ ) ] , if κ 4 ≤ n 5 , [ 1 2436 n 9 / 8 1 + κ , 1 1218 n 9 / 8 1 + κ ] , if κ 4 > n 5 . (56) \beta_n(\kappa) \in\begin{cases}
\left[\frac{1}{8(1+\kappa)},\frac{1}{4(1+\kappa)}\right], & \text{if } \kappa^4 \le n^5,\\
\left[\frac{1}{8192n^{3/2}},\frac{1}{4096n^{3/2}}\right], & \text{if } \kappa^4 > n^5
\end{cases}
\qquad
\delta_n(\kappa) \in\begin{cases}
\left[\frac{1}{32\sqrt{n}(1+\kappa)},\frac{1}{16\sqrt{n}(1+\kappa)}\right], & \text{if } \kappa^4 \le n^5,\\
\left[\frac{1}{2436n^{9/8}\sqrt{1+\kappa}},\frac{1}{1218n^{9/8}\sqrt{1+\kappa}}\right], & \text{if } \kappa^4 > n^5.
\end{cases} \tag*{(56)} β n ( κ ) ∈ { [ 8 ( 1 + κ ) 1 , 4 ( 1 + κ ) 1 ] , [ 8192 n 3/2 1 , 4096 n 3/2 1 ] , if κ 4 ≤ n 5 , if κ 4 > n 5 δ n ( κ ) ∈ ⎩ ⎨ ⎧ [ 32 n ( 1 + κ ) 1 , 16 n ( 1 + κ ) 1 ] , [ 2436 n 9/8 1 + κ 1 , 1218 n 9/8 1 + κ 1 ] , if κ 4 ≤ n 5 , if κ 4 > n 5 . (56) We separate the iterations for a fixed promise κ \kappa κ ( M , q ) (M,q) ( M , q ) ( x 0 , s 0 , μ 0 ) (x^0,s^0,\mu_0) ( x 0 , s 0 , μ 0 ) κ ≥ 1 \kappa\ge1 κ ≥ 1 κ ^ ( M ) \hat{\kappa}(M) κ ^ ( M ) β n ( κ ) \beta_n(\kappa) β n ( κ ) δ n ( κ ) \delta_n(\kappa) δ n ( κ )
( x ^ , s , μ ) ∈ N 2 ( β n ( κ ) ) (\hat{x},s,\mu) \in\mathcal{N}^2(\beta_n(\kappa)) ( x ^ , s , μ ) ∈ N 2 ( β n ( κ )) μ ≤ μ 1 \mu\le\mu_1 μ ≤ μ 1
κ ^ ( M ) > κ \hat{\kappa}(M) > \kappa κ ^ ( M ) > κ
M ∉ P ⋆ M \notin\mathcal{P}_{\star} M ∈ / P ⋆
Algorithm 3 takes a rational input ( M , q ) (M,q) ( M , q ) Theorem 2.1 , if M ∈ P ⋆ M \in\mathcal{P}_{\star} M ∈ P ⋆ F ( M , q ) ≠ ∅ \mathcal{F}(M,q) \ne\varnothing F ( M , q ) = ∅ LCP ( M , q ) \operatorname{LCP}(M,q) LCP ( M , q ) M x + q = s Mx+q=s M x + q = s x , s ≥ 0 n x,s \ge0_n x , s ≥ 0 n M ∉ P ⋆ M \notin\mathcal{P}_{\star} M ∈ / P ⋆
We let L = length ( M , q ) L=\operatorname{length}(M,q) L = length ( M , q ) Δ = 2 O ( L ) \Delta=2^{O(L)} Δ = 2 O ( L ) Lemma 8.3 . We start with the guess κ = 1 \kappa=1 κ = 1 κ \kappa κ ( M ^ , q ^ ) (\widehat{M},\widehat{q}) ( M , q ) LCP ( M ^ , q ^ ) \operatorname{LCP}(\widehat{M},\widehat{q}) LCP ( M , q ) ( x ^ 0 , s ^ 0 , μ ^ 0 ) (\widehat{x}^{0},\widehat{s}^{0},\widehat{\mu}_0) ( x 0 , s 0 , μ 0 ) Lemma 8.4 , and μ 1 \mu_1 μ 1 Lemma 8.5 .
If it returns M ^ ∉ P ⋆ \widehat{M} \notin\mathcal{P}_{\star} M ∈ / P ⋆ κ ^ ( M ^ ) > κ \hat{\kappa}(\widehat{M})>\kappa κ ^ ( M ) > κ κ \kappa κ ( ( x , x ~ ) , ( s , s ~ ) , μ ) ∈ N 2 ( β 2 n ( κ ) ) ((x,\widetilde{x}),(s,\widetilde{s}),\mu) \in\mathcal{N}^2(\beta_{2n}(\kappa)) (( x , x ) , ( s , s ) , μ ) ∈ N 2 ( β 2 n ( κ )) ( I , J ) (I,J) ( I , J ) Lemma 8.5 , and solve the LP M ^ x ‾ + q ^ = s ‾ \widehat{M}\overline{x}+\widehat{q}=\overline{s} M x + q = s x ‾ , s ‾ ≥ 0 n \overline{x},\overline{s}\ge0_n x , s ≥ 0 n x ‾ I = 0 I \overline{x}_I=0_I x I = 0 I s ‾ J = 0 J \overline{s}_J=0_J s J = 0 J ( x ‾ , s ‾ ) (\overline{x},\overline{s}) ( x , s ) κ ^ ( M ) > κ \hat{\kappa}(M)>\kappa κ ^ ( M ) > κ κ \kappa κ T T T Theorem 8.6 , then we terminate by concluding again M ∉ P ⋆ M\notin\mathcal{P}_{\star} M ∈ / P ⋆
Now we can prove our main theorem Theorem 1.2.
Theorem 1.2. Let M ∈ P ⋆ ∩ Q n × n M \in\mathcal{P}_{\star}\cap\mathbb{Q}^{n\times n} M ∈ P ⋆ ∩ Q n × n q ∈ Q n q\in\mathbb{Q}^n q ∈ Q n L L L LCP ( M , q ) \operatorname{LCP}(M,q) LCP ( M , q )
O ( min { n ( 1 + κ ^ ( M ) ) , n 9 / 8 1 + κ ^ ( M ) } ⋅ ( L + log ( 2 + κ ^ ( M ) ) ) ) O\left(\min\left\{\sqrt{n}(1+\hat{\kappa}(M)),n^{9/8}\sqrt{1+\hat{\kappa}(M)}\right\}\cdot\left(L+\log(2+\hat{\kappa}(M))\right)\right) O ( min { n ( 1 + κ ^ ( M )) , n 9/8 1 + κ ^ ( M ) } ⋅ ( L + log ( 2 + κ ^ ( M )) ) ) many linear system solves, and in addition the running time of O ( log ( 2 + κ ^ ( M ) ) ) O(\log(2+\hat{\kappa}(M))) O ( log ( 2 + κ ^ ( M ))) M x + q = s Mx+q=s M x + q = s x , s ≥ 0 n x,s\ge0_n x , s ≥ 0 n x I = 0 I x_I=0_I x I = 0 I s J = 0 J s_J=0_J s J = 0 J I , J ⊆ [ n ] I,J\subseteq[n] I , J ⊆ [ n ]
Proof. In Algorithm 3, we first check the feasibility of LCP ( M , q ) \operatorname{LCP}(M,q) LCP ( M , q ) M x + q = s Mx+q=s M x + q = s x , s ≥ 0 n x,s\ge0_n x , s ≥ 0 n LCP ( M , q ) \operatorname{LCP}(M,q) LCP ( M , q ) M ∉ P ⋆ M\notin\mathcal{P}_{\star} M ∈ / P ⋆ Theorem 2.1 .
Let us start with analyzing the subroutine FIXEDKAPPANEWTON. This returns M ∉ P ⋆ M\notin\mathcal{P}_{\star} M ∈ / P ⋆ Corollary 2.5 . By the choice of β \beta β δ \delta δ (56) , Theorem 4.2 and Theorem 6.4 guarantee that as long as κ ^ ( M ) ≤ κ \hat{\kappa}(M)\le\kappa κ ^ ( M ) ≤ κ
O ( min { n ( 1 + κ ) , n 9 / 8 1 + κ } ⋅ log ( μ 0 μ 1 ) ) (57) O\left(\min\left\{\sqrt{n}(1+\kappa),n^{9/8}\sqrt{1+\kappa}\right\}\cdot\log\left(\frac{\mu_0}{\mu_1}\right)\right) \tag*{(57)} O ( min { n ( 1 + κ ) , n 9/8 1 + κ } ⋅ log ( μ 1 μ 0 ) ) (57) Algorithm 3: Full version Newton’s Method for LCP path following
Input : M ∈ Q n × n M \in\mathbb{Q}^{n \times n} M ∈ Q n × n q ∈ Q n q \in\mathbb{Q}^{n} q ∈ Q n Output: Complementary solution ( x , s ) (x,s) ( x , s ) ( M , q ) (M,q) ( M , q ) ( M , q ) (M,q) ( M , q ) M ∉ P ⋆ M \notin\mathcal{P}_{\star} M ∈ / P ⋆
Function FixedKappaNewton( M , q , x 0 , s 0 , μ 0 , μ 1 , κ ) (M,q,x^0,s^0,\mu_0,\mu_1,\kappa) ( M , q , x 0 , s 0 , μ 0 , μ 1 , κ )
( x , s , μ ) ← ( x 0 , s 0 , μ 0 ) (x,s,\mu) \leftarrow(x^0,s^0,\mu_0) ( x , s , μ ) ← ( x 0 , s 0 , μ 0 )
β ← β n ( κ ) \beta\leftarrow\beta_n(\kappa) β ← β n ( κ ) δ ← δ n ( κ ) \delta\leftarrow\delta_n(\kappa) δ ← δ n ( κ ) (56) ;
while μ > μ 1 \mu> \mu_1 μ > μ 1 do
Compute Newton step ( Δ x , Δ s ) (\Delta x,\Delta s) ( Δ x , Δ s ) ( x , s , μ 1 + δ ) (x,s,\frac{\mu}{1+\delta}) ( x , s , 1 + δ μ ) (11) ;
if (11) has no solution or non-unique solution then return M ∉ P ⋆ M \notin\mathcal{P}_{\star} M ∈ / P ⋆
( x ′ , s ′ ) ← ( x + Δ x , s + Δ s ) (x',s') \leftarrow(x+\Delta x,s+\Delta s) ( x ′ , s ′ ) ← ( x + Δ x , s + Δ s )
if ( x ′ , s ′ ) ∉ F > 0 (x',s') \notin\mathcal{F}_{>0} ( x ′ , s ′ ) ∈ / F > 0 then return κ ^ ( M ) > κ \widehat{\kappa}(M)>\kappa κ ( M ) > κ
Compute Newton step ( Δ x , Δ s ) (\Delta x,\Delta s) ( Δ x , Δ s ) ( x ′ , s ′ , μ 1 + δ ) (x',s',\frac{\mu}{1+\delta}) ( x ′ , s ′ , 1 + δ μ ) (11) ;
if (11) has no solution or non-unique solution then return M ∉ P ⋆ M \notin\mathcal{P}_{\star} M ∈ / P ⋆
( x g , s g , μ g ) ← ( x ′ + Δ x , s ′ + Δ s , μ 1 + δ ) (x_g,s_g,\mu_g) \leftarrow(x'+\Delta x,s'+\Delta s,\frac{\mu}{1+\delta}) ( x g , s g , μ g ) ← ( x ′ + Δ x , s ′ + Δ s , 1 + δ μ )
if ( x g , s g , μ g ) ∉ N 2 ( β ) (x_g,s_g,\mu_g) \notin\mathcal{N}^{2}(\beta) ( x g , s g , μ g ) ∈ / N 2 ( β ) then return κ ^ ( M ) > κ \widehat{\kappa}(M)>\kappa κ ( M ) > κ
( x , s , μ ) ← ( x g , s g , μ g ) (x,s,\mu) \leftarrow(x_g,s_g,\mu_g) ( x , s , μ ) ← ( x g , s g , μ g )
return ( x , s , μ ) (x,s,\mu) ( x , s , μ )
U ← LPSolve ( M x + q = s , x , s ≥ 0 n ) U \leftarrow\operatorname{LPSolve}(Mx+q=s,\ x,s\geq0_n) U ← LPSolve ( M x + q = s , x , s ≥ 0 n )
if U = I n f e a s i b l e U=Infeasible U = I n f e a s ib l e then return I n f e a s i b l e Infeasible I n f e a s ib l e
Δ ← 2 C 1 length ( M , q ) \Delta\leftarrow2^{C_1\operatorname{length}(M,q)} Δ ← 2 C 1 length ( M , q ) C 1 C_1 C 1
T ← 2 C 2 n length ( M ) T \leftarrow2^{C_2n\operatorname{length}(M)} T ← 2 C 2 n length ( M ) C 2 C_2 C 2
( M ^ , q ^ ) (\widehat{M},\widehat{q}) ( M , q ) ( M ^ , q ^ ) (\widehat{M},\widehat{q}) ( M , q ) q ~ \widetilde{q} q (54) ;
( x ^ 0 , s ^ 0 , μ ^ 0 ) (\widehat{x}^{0},\widehat{s}^{0},\widehat{\mu}_0) ( x 0 , s 0 , μ 0 ) (54) ;
κ ← 1 \kappa\leftarrow1 κ ← 1
while κ ≤ T \kappa\leq T κ ≤ T do
μ 1 ← 1 / ( 16384 n 3 Δ 2 ( κ + 1 ) 2 ) \mu_1 \leftarrow1/(16384n^3\Delta^2(\kappa+1)^2) μ 1 ← 1/ ( 16384 n 3 Δ 2 ( κ + 1 ) 2 )
R ← R \leftarrow R ← ( M ^ , q ^ , x ^ 0 , s ^ 0 , μ ^ 0 , μ 1 , κ ) (\widehat{M},\widehat{q},\widehat{x}^{0},\widehat{s}^{0},\widehat{\mu}_0,\mu_1,\kappa) ( M , q , x 0 , s 0 , μ 0 , μ 1 , κ )
if R = ( M ^ ∉ P ⋆ ) R=(\widehat{M}\notin\mathcal{P}_{\star}) R = ( M ∈ / P ⋆ ) then return M ∉ P ⋆ M\notin\mathcal{P}_{\star} M ∈ / P ⋆
if R = ( ( x , x ~ ) , ( s , s ~ ) , μ ) R=((x,\widetilde{x}),(s,\widetilde{s}),\mu) R = (( x , x ) , ( s , s ) , μ ) then
I ← { i ∈ [ n ] : x i ≤ 2 n μ } I\leftarrow\{i\in[n]:x_i\leq\sqrt{2n\mu}\} I ← { i ∈ [ n ] : x i ≤ 2 n μ } J ← { i ∈ [ n ] : s i ≤ 2 n μ } J\leftarrow\{i\in[n]:s_i\leq\sqrt{2n\mu}\} J ← { i ∈ [ n ] : s i ≤ 2 n μ }
U ← LPSolve ( M x + q = s , x , s ≥ 0 n , x I = 0 I , s J = 0 J ) U\leftarrow\operatorname{LPSolve}(Mx+q=s,\ x,s\geq0_n,\ x_I=0_I,\ s_J=0_J) U ← LPSolve ( M x + q = s , x , s ≥ 0 n , x I = 0 I , s J = 0 J )
if U = ( x ′ , s ′ ) U=(x',s') U = ( x ′ , s ′ ) then return ( x ′ , s ′ ) (x',s') ( x ′ , s ′ )
κ ← 2 κ \kappa\leftarrow2\kappa κ ← 2 κ
return M ∉ P ⋆ M\notin\mathcal{P}_{\star} M ∈ / P ⋆
iterations. Note that Theorem 4.2 analyzes Algorithm 1 that only does a single Newton step whereas here we perform two Newton steps; but this does not increase the asymptotic running time.
Consider now the cases when the subroutine returns κ ^ ( M ) > κ \widehat{\kappa}(M)>\kappa κ ( M ) > κ ( x ′ , s ′ ) (x',s') ( x ′ , s ′ ) F > 0 \mathcal{F}_{>0} F > 0 ( x g , s g , μ g ) (x_g,s_g,\mu_g) ( x g , s g , μ g ) N 2 ( β ) \mathcal{N}^{2}(\beta) N 2 ( β ) ( x ′ , s ′ ) ∈ F > 0 (x',s')\in\mathcal{F}_{>0} ( x ′ , s ′ ) ∈ F > 0 ( x g , s g , μ g ) ∈ N 2 ( β ) (x_g,s_g,\mu_g)\in\mathcal{N}^{2}(\beta) ( x g , s g , μ g ) ∈ N 2 ( β ) M ∈ P ⋆ ( κ ) M\in\mathcal{P}_{\star}(\kappa) M ∈ P ⋆ ( κ ) N R ( β ) ⊆ N ( β ) \mathcal{N}^{R}(\beta)\subseteq\mathcal{N}(\beta) N R ( β ) ⊆ N ( β ) (13) . From the failure of the analysis, it follows that the assumption M ∈ P ⋆ ( κ ) M\in\mathcal{P}_{\star}(\kappa) M ∈ P ⋆ ( κ ) κ ^ ( M ) > κ \widehat{\kappa}(M)>\kappa κ ( M ) > κ κ 4 > n 5 \kappa^4>n^5 κ 4 > n 5 N R ( β ) \mathcal{N}^{R}(\beta) N R ( β ) N ( β ) \mathcal{N}(\beta) N ( β ) μ 1 \mu_1 μ 1 M ∉ P ⋆ ( κ ) M\notin\mathcal{P}_{\star}(\kappa) M ∈ / P ⋆ ( κ )
Consider now the embedded system LCP( M ^ , q ^ ) (\widehat{M},\widehat{q}) ( M , q ) ( x ^ 0 , s ^ 0 , μ ^ 0 ) (\widehat{x}^{0},\widehat{s}^{0},\widehat{\mu}_0) ( x 0 , s 0 , μ 0 ) μ 1 \mu_1 μ 1 κ ^ ( M ^ ) = κ ^ ( M ) \widehat{\kappa}(\widehat{M})=\widehat{\kappa}(M) κ ( M ) = κ ( M ) ( M ^ , q ^ , x ^ 0 , s ^ 0 , μ ^ 0 , μ 1 , κ ) (\widehat{M},\widehat{q},\widehat{x}^{0},\widehat{s}^{0},\widehat{\mu}_0,\mu_1,\kappa) ( M , q , x 0 , s 0 , μ 0 , μ 1 , κ ) M ^ ∉ P ⋆ \widehat{M}\notin\mathcal{P}_{\star} M ∈ / P ⋆ κ ^ ( M ) > κ \widehat{\kappa}(M)>\kappa κ ( M ) > κ M M M log ( μ 0 / μ 1 ) = O ( L + log ( 2 + κ ) ) \log(\mu_0/\mu_1)=O(L+\log(2+\kappa)) log ( μ 0 / μ 1 ) = O ( L + log ( 2 + κ )) ( M ^ , q ^ , x ^ 0 , s ^ 0 , μ ^ 0 , μ 1 , κ ) (\widehat{M},\widehat{q},\widehat{x}^{0},\widehat{s}^{0},\widehat{\mu}_0,\mu_1,\kappa) ( M , q , x 0 , s 0 , μ 0 , μ 1 , κ ) ( x , x ~ ) , ( s , s ~ ) , μ ∈ N 2 ( β 2 n ( κ ) ) (x,\widetilde{x}),(s,\widetilde{s}),\mu\in\mathcal{N}^{2}(\beta_{2n}(\kappa)) ( x , x ) , ( s , s ) , μ ∈ N 2 ( β 2 n ( κ )) μ ≤ μ 1 \mu\leq\mu_1 μ ≤ μ 1 κ ^ ( M ) > κ \widehat{\kappa}(M)>\kappa κ ( M ) > κ ( x ′ , s ′ ) (x',s') ( x ′ , s ′ ) L C P ( M , q ) \mathrm{LCP}(M,q) LCP ( M , q ) x I ′ = 0 I x'_I = 0_I x I ′ = 0 I s J ′ = 0 J s'_J = 0_J s J ′ = 0 J I I I J J J κ \kappa κ
Since κ \kappa κ (57) with log ( μ 0 / μ 1 ) = O ( L + log ( 2 + κ ) ) \log(\mu_0/\mu_1) = O(L + \log(2 + \kappa)) log ( μ 0 / μ 1 ) = O ( L + log ( 2 + κ )) κ ≤ 2 κ ^ ( M ) + 1 \kappa\le2\hat{\kappa}(M) + 1 κ ≤ 2 κ ^ ( M ) + 1 O ( log ( 2 + κ ^ ( M ) ) ) O(\log(2 + \hat{\kappa}(M))) O ( log ( 2 + κ ^ ( M )))
Deferred Proofs
Proof of Theorem 1.1
First we restate the theorem.
Theorem 1.1. Let M ∈ R n × n M \in\mathbb{R}^{n \times n} M ∈ R n × n M ∈ P ⋆ M \in\mathcal{P}_{\star} M ∈ P ⋆ I n + D M D I_n + DMD I n + D M D D ∈ D > 0 n D \in\mathcal{D}_{>0}^{n} D ∈ D > 0 n
sup D ∈ D > 0 n ∥ ( I n + D M D ) − 1 ∥ 2 \sup_{D\in\mathcal{D}_{>0}^{n}} \left\|(I_n + DMD)^{-1}\right\|_2 D ∈ D > 0 n sup ( I n + D M D ) − 1 2 is finite. If M ∈ P ⋆ M \in\mathcal{P}_{\star} M ∈ P ⋆
sup D ∈ D > 0 n ∥ ( I n + D M D ) − 1 ∥ 2 = 1 + κ ^ ( M ) 2 1 / 4 + κ ^ ( M ) . (58) \sup_{D\in\mathcal{D}_{>0}^{n}} \left\|(I_n + DMD)^{-1}\right\|_2 = \sqrt{1 + \frac{\hat{\kappa}(M)^2}{1/4 + \hat{\kappa}(M)}}. \tag*{(58)} D ∈ D > 0 n sup ( I n + D M D ) − 1 2 = 1 + 1/4 + κ ^ ( M ) κ ^ ( M ) 2 . (58) The ≤ \le ≤ Corollary 9.3 , and the ≥ \ge ≥ Lemma 9.4 .
Proposition 9.1. Let κ ≥ 0 \kappa\ge0 κ ≥ 0 P , N ≥ 0 P, N \ge0 P , N ≥ 0 ( 1 + 4 κ ) P − N ≥ 0 (1 + 4\kappa)P - N \ge0 ( 1 + 4 κ ) P − N ≥ 0 P − N ≥ − 2 κ 1 + 2 κ ( P + N ) P - N \ge-\frac{2\kappa}{1+2\kappa}(P + N) P − N ≥ − 1 + 2 κ 2 κ ( P + N )
Proof. If κ = 0 \kappa= 0 κ = 0 κ > 0 \kappa> 0 κ > 0 α = 2 κ 1 + 2 κ \alpha= \frac{2\kappa}{1+2\kappa} α = 1 + 2 κ 2 κ 1 + α 1 − α = 1 + 2 α 1 − α = 1 + 2 ⋅ 2 κ / ( 1 + 2 κ ) 1 − 2 κ / ( 1 + 2 κ ) = 1 + 4 κ \frac{1+\alpha}{1-\alpha} = 1 + \frac{2\alpha}{1-\alpha} = 1 + \frac{2\cdot2\kappa/(1+2\kappa)}{1-2\kappa/(1+2\kappa)} = 1 + 4\kappa 1 − α 1 + α = 1 + 1 − α 2 α = 1 + 1 − 2 κ / ( 1 + 2 κ ) 2 ⋅ 2 κ / ( 1 + 2 κ ) = 1 + 4 κ ( 1 + 4 κ ) P − N ≥ 0 (1 + 4\kappa)P - N \ge0 ( 1 + 4 κ ) P − N ≥ 0
1 + α 1 − α P − N ≥ 0. \frac{1+\alpha}{1-\alpha}P - N \ge0. 1 − α 1 + α P − N ≥ 0. By re-arranging terms, P − N ≥ − α ( P + N ) P - N \ge-\alpha(P + N) P − N ≥ − α ( P + N ) P − N ≥ − 2 κ 1 + 2 κ ( P + N ) P - N \ge-\frac{2\kappa}{1+2\kappa}(P + N) P − N ≥ − 1 + 2 κ 2 κ ( P + N )
Lemma 9.2. If M ∈ P ⋆ ( κ ) ∩ R n × n M \in\mathcal{P}_{\star}(\kappa) \cap\mathbb{R}^{n \times n} M ∈ P ⋆ ( κ ) ∩ R n × n
M + M T ⪰ − ( 2 κ 1 + 2 κ ) 2 ⋅ I n − M T M . M + M^{\mathsf{T}} \succeq-\left(\frac{2\kappa}{1+2\kappa}\right)^2 \cdot I_n - M^{\mathsf{T}}M. M + M T ⪰ − ( 1 + 2 κ 2 κ ) 2 ⋅ I n − M T M . Proof. If κ = 0 \kappa= 0 κ = 0 κ > 0 \kappa> 0 κ > 0 x ∈ R n x \in\mathbb{R}^n x ∈ R n P = ∑ i : x i ( M x ) i ≥ 0 x i ( M x ) i P = \sum_{i:x_i(Mx)_i\ge0} x_i(Mx)_i P = ∑ i : x i ( M x ) i ≥ 0 x i ( M x ) i N = − ∑ i : x i ( M x ) i < 0 x i ( M x ) i N = -\sum_{i:x_i(Mx)_i<0} x_i(Mx)_i N = − ∑ i : x i ( M x ) i < 0 x i ( M x ) i ( 1 + 4 κ ) P − N ≥ 0 (1 + 4\kappa)P - N \ge0 ( 1 + 4 κ ) P − N ≥ 0 Proposition 9.1 , we have
P − N ≥ − 2 κ 1 + 2 κ ( P + N ) , P - N \ge-\frac{2\kappa}{1+2\kappa}(P + N), P − N ≥ − 1 + 2 κ 2 κ ( P + N ) , where x T ( M + M T ) x = 2 x T M x = 2 ( P − N ) x^{\mathsf{T}}(M + M^{\mathsf{T}})x = 2x^{\mathsf{T}}Mx = 2(P - N) x T ( M + M T ) x = 2 x T M x = 2 ( P − N )
P + N = ∑ i = 1 n ∣ x i ∣ ∣ M x i ∣ ≤ ∑ i = 1 n 1 2 ( 2 κ 1 + 2 κ ⋅ x i 2 + 1 + 2 κ 2 κ ⋅ ( M x ) i 2 ) = 1 2 ( 2 κ 1 + 2 κ ⋅ ∥ x ∥ 2 + 1 + 2 κ 2 κ ⋅ ∥ M x ∥ 2 ) . P + N = \sum_{i=1}^{n}|x_i||Mx_i| \le\sum_{i=1}^{n}\frac{1}{2}\left(\frac{2\kappa}{1+2\kappa}\cdot x_i^2 + \frac{1+2\kappa}{2\kappa}\cdot(Mx)_i^2\right)
= \frac{1}{2}\left(\frac{2\kappa}{1+2\kappa}\cdot\|x\|^2 + \frac{1+2\kappa}{2\kappa}\cdot\|Mx\|^2\right). P + N = i = 1 ∑ n ∣ x i ∣∣ M x i ∣ ≤ i = 1 ∑ n 2 1 ( 1 + 2 κ 2 κ ⋅ x i 2 + 2 κ 1 + 2 κ ⋅ ( M x ) i 2 ) = 2 1 ( 1 + 2 κ 2 κ ⋅ ∥ x ∥ 2 + 2 κ 1 + 2 κ ⋅ ∥ M x ∥ 2 ) . Combining the two inequalities yields
x T ( M + M T ) x ≥ − ( 2 κ 1 + 2 κ ) 2 x T x − x T M T M x ∀ x ∈ R n , x^{\mathsf{T}}(M + M^{\mathsf{T}})x \ge-\left(\frac{2\kappa}{1+2\kappa}\right)^2 x^{\mathsf{T}}x - x^{\mathsf{T}}M^{\mathsf{T}}Mx \quad\forall x \in\mathbb{R}^n, x T ( M + M T ) x ≥ − ( 1 + 2 κ 2 κ ) 2 x T x − x T M T M x ∀ x ∈ R n , proving the claim.
Corollary 9.3. If M ∈ P ⋆ ( κ ) ∩ R n × n M \in\mathcal{P}_{\star}(\kappa) \cap\mathbb{R}^{n \times n} M ∈ P ⋆ ( κ ) ∩ R n × n D ∈ D > 0 n D \in\mathcal{D}_{>0}^{n} D ∈ D > 0 n
∥ ( I n + D M D ) − 1 ∥ 2 ≤ 1 + κ 2 1 / 4 + κ . \left\|(I_n + D M D)^{-1}\right\|_2 \leq\sqrt{1 + \frac{\kappa^2}{1/4 + \kappa}}. ( I n + D M D ) − 1 2 ≤ 1 + 1/4 + κ κ 2 . Proof. Note that by Lemma 2.2, M ∈ P ⋆ ( κ ) M \in\mathcal{P}_{\star}(\kappa) M ∈ P ⋆ ( κ ) D M D ∈ P ⋆ ( κ ) D M D \in\mathcal{P}_{\star}(\kappa) D M D ∈ P ⋆ ( κ ) D = I n D = I_n D = I n ∥ ( I n + M ) − 1 ∥ 2 ≤ 1 + κ 2 1 / 4 + κ \left\|(I_n + M)^{-1}\right\|_2 \leq\sqrt{1 + \frac{\kappa^2}{1/4+\kappa}} ( I n + M ) − 1 2 ≤ 1 + 1/4 + κ κ 2 Lemma 9.2 we can lower bound
( I n + M ) T ( I n + M ) = I n + M T M + M + M T ⪰ I n + M T M − ( 2 κ 1 + 2 κ ) 2 ⋅ I n − M T M = 1 + 4 κ ( 1 + 2 κ ) 2 ⋅ I n . (I_n + M)^{\mathsf{T}}(I_n + M) = I_n + M^{\mathsf{T}}M + M + M^{\mathsf{T}} \succeq I_n + M^{\mathsf{T}}M - \left(\frac{2\kappa}{1+2\kappa}\right)^2 \cdot I_n - M^{\mathsf{T}}M = \frac{1+4\kappa}{(1+2\kappa)^2} \cdot I_n . ( I n + M ) T ( I n + M ) = I n + M T M + M + M T ⪰ I n + M T M − ( 1 + 2 κ 2 κ ) 2 ⋅ I n − M T M = ( 1 + 2 κ ) 2 1 + 4 κ ⋅ I n . Equivalently we have
∥ ( I n + M ) − 1 ∥ 2 ≤ 1 + κ 2 1 / 4 + κ . \left\|(I_n + M)^{-1}\right\|_2 \leq\sqrt{1 + \frac{\kappa^2}{1/4+\kappa}}. ( I n + M ) − 1 2 ≤ 1 + 1/4 + κ κ 2 . which is what we needed.
We now show the converse direction, completing the proof of Theorem 1.1.
Lemma 9.4. Suppose that for M ∈ R n × n M \in\mathbb{R}^{n \times n} M ∈ R n × n κ ≥ 0 \kappa\geq0 κ ≥ 0 D ∈ D > 0 n D \in\mathcal{D}_{>0}^{n} D ∈ D > 0 n
∥ ( I n + D M D ) − 1 ∥ 2 ≤ 1 + κ 2 1 / 4 + κ , \left\|(I_n + D M D)^{-1}\right\|_2 \leq\sqrt{1 + \frac{\kappa^2}{1/4+\kappa}}, ( I n + D M D ) − 1 2 ≤ 1 + 1/4 + κ κ 2 , then M M M P ⋆ ( κ ) \mathcal{P}_{\star}(\kappa) P ⋆ ( κ ) κ ^ ( M ) ≤ κ \hat{\kappa}(M) \leq\kappa κ ^ ( M ) ≤ κ
Proof. Define ε = 1 / ( 1 + κ 2 1 / 4 + κ ) = 1 + 4 κ 1 + 4 κ + 4 κ 2 \varepsilon= 1/\left(1+\frac{\kappa^2}{1/4+\kappa}\right) = \frac{1+4\kappa}{1+4\kappa+4\kappa^2} ε = 1/ ( 1 + 1/4 + κ κ 2 ) = 1 + 4 κ + 4 κ 2 1 + 4 κ ∥ ( I n + D M D ) − 1 ∥ 2 ≤ 1 / ε \left\|(I_n + D M D)^{-1}\right\|_2 \leq\sqrt{1/\varepsilon} ( I n + D M D ) − 1 2 ≤ 1/ ε
( 1 − ε ) I n + D ( M + M T ) D + D M T D 2 M D ⪰ 0 (1-\varepsilon)I_n + D(M+M^{\mathsf{T}})D + D M^{\mathsf{T}}D^2 M D \succeq0 ( 1 − ε ) I n + D ( M + M T ) D + D M T D 2 M D ⪰ 0 and again equivalently as
( 1 − ε ) D − 2 + M T D 2 M + M + M T ⪰ 0. (1-\varepsilon)D^{-2} + M^{\mathsf{T}}D^2M + M + M^{\mathsf{T}} \succeq0. ( 1 − ε ) D − 2 + M T D 2 M + M + M T ⪰ 0. Thus for any x ∈ R n x \in\mathbb{R}^n x ∈ R n D ∈ D > 0 n D \in\mathcal{D}_{>0}^{n} D ∈ D > 0 n
∑ i = 1 n ( ( 1 − ε ) x i 2 d i 2 + ( M x ) i 2 ⋅ d i 2 ) + 2 x T M x ≥ 0. (59) \sum_{i=1}^{n}\left((1-\varepsilon)\frac{x_i^2}{d_i^2} + (Mx)_i^2 \cdot d_i^2\right) + 2x^{\mathsf{T}}Mx \geq0 . \tag*{(59)} i = 1 ∑ n ( ( 1 − ε ) d i 2 x i 2 + ( M x ) i 2 ⋅ d i 2 ) + 2 x T M x ≥ 0. (59) Now we take the infimum over all strictly positive d i d_i d i a , b ≥ 0 a,b \geq0 a , b ≥ 0 inf t > 0 ( a t + b t ) = 2 a b \inf_{t>0}\left(\frac{a}{t}+bt\right)=2\sqrt{ab} inf t > 0 ( t a + b t ) = 2 ab a , b > 0 a,b>0 a , b > 0 t = a / b t=\sqrt{a/b} t = a / b a = 0 a=0 a = 0 t → 0 t\to0 t → 0 b = 0 b=0 b = 0 t → ∞ t\to\infty t → ∞ a = b = 0 a=b=0 a = b = 0 t t t
Therefore, applying this coordinatewise with t = d i 2 t=d_i^2 t = d i 2 a = ( 1 − ε ) x i 2 a=(1-\varepsilon)x_i^2 a = ( 1 − ε ) x i 2 b = ( M x ) i 2 b=(Mx)_i^2 b = ( M x ) i 2 (59) we obtain that
1 − ε ∑ i = 1 n ∣ x i ∣ ∣ M x ∣ i + x T M x ≥ 0 \sqrt{1-\varepsilon}\sum_{i=1}^{n}|x_i||Mx|_i + x^{\mathsf{T}}Mx \geq0 1 − ε i = 1 ∑ n ∣ x i ∣∣ M x ∣ i + x T M x ≥ 0 Again letting P = ∑ i : x i ( M x ) i ≥ 0 x i ( M x ) i P=\sum_{i:x_i(Mx)_i\geq0}x_i(Mx)_i P = ∑ i : x i ( M x ) i ≥ 0 x i ( M x ) i N = − ∑ i : x i ( M x ) i < 0 x i ( M x ) i N=-\sum_{i:x_i(Mx)_i<0}x_i(Mx)_i N = − ∑ i : x i ( M x ) i < 0 x i ( M x ) i
( P − N ) ≥ − 1 − ε ( P + N ) . (P-N) \geq-\sqrt{1-\varepsilon}(P+N). ( P − N ) ≥ − 1 − ε ( P + N ) . By Proposition 9.1 , we equivalently have that ( 1 + 4 κ ~ ) P − N ≥ 0 (1+4\widetilde{\kappa})P-N\geq0 ( 1 + 4 κ ) P − N ≥ 0 κ ~ \widetilde{\kappa} κ 1 − ε = 2 κ ~ 1 + 2 κ ~ = 1 − 1 1 + 2 κ ~ \sqrt{1-\varepsilon}=\frac{2\widetilde{\kappa}}{1+2\widetilde{\kappa}}=1-\frac{1}{1+2\widetilde{\kappa}} 1 − ε = 1 + 2 κ 2 κ = 1 − 1 + 2 κ 1 2 κ ~ = 1 1 − 1 − ε − 1 2\widetilde{\kappa}=\frac{1}{1-\sqrt{1-\varepsilon}}-1 2 κ = 1 − 1 − ε 1 − 1 ε \varepsilon ε
2 κ ~ = 1 1 − 4 κ 2 ( 1 + 2 κ ) 2 − 1 = 2 κ , 2\widetilde{\kappa}=\frac{1}{1-\sqrt{\frac{4\kappa^2}{(1+2\kappa)^2}}}-1=2\kappa, 2 κ = 1 − ( 1 + 2 κ ) 2 4 κ 2 1 − 1 = 2 κ , which shows that M M M P ⋆ ( κ ) \mathcal{P}_{\star}(\kappa) P ⋆ ( κ )
Proof of Theorem 6.4
Theorem 6.4. Let M ∈ P ( κ ) M \in\mathcal{P}(\kappa) M ∈ P ( κ ) β ∈ [ 1 8192 n 3 / 2 , 1 4096 n 3 / 2 ] \beta\in\left[\frac{1}{8192n^{3/2}},\frac{1}{4096n^{3/2}}\right] β ∈ [ 8192 n 3/2 1 , 4096 n 3/2 1 ] δ ∈ [ 1 2436 n 9 / 8 1 + κ , 1 1218 n 9 / 8 1 + κ ] \delta\in\left[\frac{1}{2436n^{9/8}\sqrt{1+\kappa}},\frac{1}{1218n^{9/8}\sqrt{1+\kappa}}\right] δ ∈ [ 2436 n 9/8 1 + κ 1 , 1218 n 9/8 1 + κ 1 ] ( x , s , μ ) ∈ N R ( β ) (x,s,\mu) \in\mathcal{N}^{R}(\beta) ( x , s , μ ) ∈ N R ( β ) μ ′ = μ / ( 1 + δ ) \mu'=\mu/(1+\delta) μ ′ = μ / ( 1 + δ ) μ ′ \mu' μ ′ (11) , produce two iterates ( x ′ , s ′ ) , ( x ′ ′ , s ′ ′ ) ∈ F > 0 (x',s'),(x'',s'') \in\mathcal{F}_{>0} ( x ′ , s ′ ) , ( x ′′ , s ′′ ) ∈ F > 0 ( x ′ ′ , s ′ ′ , μ ′ ) ∈ N R ( β ) (x'',s'',\mu') \in\mathcal{N}^{R}(\beta) ( x ′′ , s ′′ , μ ′ ) ∈ N R ( β )
Proof. We keep the same notation as in the proof of Lemma 6.1 , and recall ∥ Q H , x , μ err ( x , s , μ ) ∥ 2 ≤ β \lVert Q_{H,x,\mu}\operatorname{err}(x,s,\mu)\rVert_{2}\leq\beta ∥ Q H , x , μ err ( x , s , μ ) ∥ 2 ≤ β Δ x ‾ : = p + r \overline{\Delta x}:=p+r Δ x := p + r p : = − δ H 1 n p:=-\delta H1_n p := − δ H 1 n r : = ( 1 + δ ) H err ( x , s , μ ) r:=(1+\delta)H\operatorname{err}(x,s,\mu) r := ( 1 + δ ) H err ( x , s , μ )
Applying Lemma 5.7 to the change from μ \mu μ μ ′ \mu' μ ′ q H ≤ exp ( 8 K n δ ) q H , x , μ q_H\leq\exp(8K n\delta)q_{H,x,\mu} q H ≤ exp ( 8 K n δ ) q H , x , μ ∥ ( 1 + δ ) Q H err ( x , s , μ ) ∥ 2 ≤ exp ( 65 / 1024 ) β ≤ 3 β / 2 \lVert(1+\delta)Q_H\operatorname{err}(x,s,\mu)\rVert_2\leq\exp(65/1024)\beta\leq3\beta/2 ∥( 1 + δ ) Q H err ( x , s , μ ) ∥ 2 ≤ exp ( 65/1024 ) β ≤ 3 β /2 p = − δ H 1 n p=-\delta H1_n p = − δ H 1 n ∥ 1 n ∥ 2 = n \lVert1_n\rVert_2=\sqrt{n} ∥ 1 n ∥ 2 = n (34) and (35) give
∥ p ∥ 2 ≤ δ n ( 1 + κ ) , \lVert p\rVert_2\leq\delta\sqrt{n(1+\kappa)}, ∥ p ∥ 2 ≤ δ n ( 1 + κ ) , ∥ Q H p ∥ ∞ ≤ 32 δ n ( 1 + κ ) , \lVert Q_Hp\rVert_\infty\leq32\delta\sqrt{n(1+\kappa)}, ∥ Q H p ∥ ∞ ≤ 32 δ n ( 1 + κ ) , ∥ H diag ( p ) ∥ 2 ≤ 2 δ n 1 + κ . \lVert H\operatorname{diag}(p)\rVert_2\leq2\delta n\sqrt{1+\kappa}. ∥ H diag ( p ) ∥ 2 ≤ 2 δ n 1 + κ . For r r r ∥ Q H H Q H − 1 ∥ max ≤ K = 8 \lVert Q_HHQ_H^{-1}\rVert_{\max}\leq K=8 ∥ Q H H Q H − 1 ∥ m a x ≤ K = 8 Lemma 5.4 and q H ≥ 1 q_H\geq1 q H ≥ 1 κ \kappa κ
∥ Q H r ∥ ∞ ≤ 12 n β , \lVert Q_Hr\rVert_\infty\leq12\sqrt{n}\,\beta, ∥ Q H r ∥ ∞ ≤ 12 n β , ∥ r ∥ 2 ≤ 12 n β , \lVert r\rVert_2\leq12n\beta, ∥ r ∥ 2 ≤ 12 n β , ∥ H diag ( r ) ∥ 2 ≤ ∥ H diag ( r ) ∥ F ≤ 8 n ∥ Q H r ∥ 2 ≤ 96 n 3 / 2 β . \lVert H\operatorname{diag}(r)\rVert_2\leq\lVert H\operatorname{diag}(r)\rVert_F\leq8\sqrt{n}\lVert Q_Hr\rVert_2\leq96n^{3/2}\beta. ∥ H diag ( r ) ∥ 2 ≤ ∥ H diag ( r ) ∥ F ≤ 8 n ∥ Q H r ∥ 2 ≤ 96 n 3/2 β . These replace the use of (33) . In particular, with U = diag ( 1 n + Δ x ‾ ) U=\operatorname{diag}(1_n+\overline{\Delta x}) U = diag ( 1 n + Δ x ) δ \delta δ β \beta β
∥ Δ x ‾ ∥ 1 ≤ δ n 1 + κ + 12 n 3 / 2 β ≤ 1 256 , \lVert\overline{\Delta x}\rVert_1\leq\delta n\sqrt{1+\kappa}+12n^{3/2}\beta\leq\frac{1}{256}, ∥ Δ x ∥ 1 ≤ δ n 1 + κ + 12 n 3/2 β ≤ 256 1 , ∥ H ( U − 2 − I n ) ∥ 2 ≤ 4 ∥ H diag ( Δ x ‾ ) ∥ 2 ≤ 8 δ n 1 + κ + 384 n 3 / 2 β ≤ 13 128 < 1 2 . \lVert H(U^{-2}-I_n)\rVert_2\leq4\lVert H\operatorname{diag}(\overline{\Delta x})\rVert_2\leq8\delta n\sqrt{1+\kappa}+384n^{3/2}\beta\leq\frac{13}{128}<\frac{1}{2}. ∥ H ( U − 2 − I n ) ∥ 2 ≤ 4 ∥ H diag ( Δ x ) ∥ 2 ≤ 8 δ n 1 + κ + 384 n 3/2 β ≤ 128 13 < 2 1 . Thus x ′ , s ′ > 0 x',s'>0 x ′ , s ′ > 0 (36) , and the argument leading to (39) still gives ∥ H ′ diag ( p ) ∥ 2 ≤ 4 ∥ H diag ( p ) ∥ 2 \lVert H'\operatorname{diag}(p)\rVert_2\leq4\lVert H\operatorname{diag}(p)\rVert_2 ∥ H ′ diag ( p ) ∥ 2 ≤ 4 ∥ H diag ( p ) ∥ 2 ∣ log ( 1 + a ) ∣ ≤ 4 3 ∣ a ∣ |\log(1+a)|\leq\frac{4}{3}|a| ∣ log ( 1 + a ) ∣ ≤ 3 4 ∣ a ∣ ∣ a ∣ ≤ 1 4 |a|\leq\frac{1}{4} ∣ a ∣ ≤ 4 1 Lemma 5.7 . This gives ∥ log q H ′ − log q H ∥ ∞ ≤ 3 4 K ∥ Δ x ‾ ∥ 1 ≤ 3 2 < log 2 \lVert\log q_{H'}-\log q_H\rVert_\infty\leq\frac{3}{4}K\lVert\overline{\Delta x}\rVert_1\leq\frac{3}{2}<\log2 ∥ log q H ′ − log q H ∥ ∞ ≤ 4 3 K ∥ Δ x ∥ 1 ≤ 2 3 < log 2 q H ′ ≤ 2 q H q_{H'}\leq2q_H q H ′ ≤ 2 q H
For the second step, write Δ x ‾ ′ = z + w \overline{\Delta x}'=z+w Δ x ′ = z + w z : = H ′ p 2 z:=H'p^2 z := H ′ p 2 w : = H ′ ( 2 p r + r 2 ) w:=H'(2pr+r^2) w := H ′ ( 2 p r + r 2 ) z z z (41) and (42) :
∥ z ∥ 2 ≤ 4 ∥ H diag ( p ) ∥ 2 ∥ p ∥ 2 ≤ 8 δ 2 n 3 / 2 ( 1 + κ ) , \lVert z\rVert_2\leq4\lVert H\operatorname{diag}(p)\rVert_2\lVert p\rVert_2\leq8\delta^2n^{3/2}(1+\kappa), ∥ z ∥ 2 ≤ 4 ∥ H diag ( p ) ∥ 2 ∥ p ∥ 2 ≤ 8 δ 2 n 3/2 ( 1 + κ ) , ∥ Q H ′ z ∥ ∞ ≤ 2 K ∥ Q H p ∥ ∞ ∥ p ∥ 1 ≤ 512 δ 2 n 3 / 2 ( 1 + κ ) . \lVert Q_{H'}z\rVert_\infty\leq2K\lVert Q_Hp\rVert_\infty\lVert p\rVert_1\leq512\delta^2n^{3/2}(1+\kappa). ∥ Q H ′ z ∥ ∞ ≤ 2 K ∥ Q H p ∥ ∞ ∥ p ∥ 1 ≤ 512 δ 2 n 3/2 ( 1 + κ ) . To bound the terms involving r r r Q H ′ H ′ Q H ′ − 1 Q_{H'}H'Q_{H'}^{-1} Q H ′ H ′ Q H ′ − 1 q H ′ ≤ 2 q H q_{H'}\leq2q_H q H ′ ≤ 2 q H
∥ Q H ′ w ∥ ∞ ≤ 2 K ∥ Q H r ∥ ∞ ( 2 ∥ p ∥ 1 + ∥ r ∥ 1 ) \lVert Q_{H'}w\rVert_\infty\leq2K\lVert Q_Hr\rVert_\infty(2\lVert p\rVert_1+\lVert r\rVert_1) ∥ Q H ′ w ∥ ∞ ≤ 2 K ∥ Q H r ∥ ∞ ( 2 ∥ p ∥ 1 + ∥ r ∥ 1 ) ≤ 384 δ n 3 / 2 1 + κ β + 2304 n 2 β 2 ≤ ( 3 8 + 9 16 ) n β ≤ n β . \leq384\delta n^{3/2}\sqrt{1+\kappa}\,\beta+2304n^2\beta^2\leq\left(\frac{3}{8}+\frac{9}{16}\right)\sqrt{n}\,\beta\leq\sqrt{n}\,\beta. ≤ 384 δ n 3/2 1 + κ β + 2304 n 2 β 2 ≤ ( 8 3 + 16 9 ) n β ≤ n β . Consequently,
∥ Δ x ‾ ′ ∥ 1 ≤ n ∥ z ∥ 2 + n ∥ Q H ′ w ∥ ∞ ≤ 8 δ 2 n 2 ( 1 + κ ) + n 3 / 2 β ≤ 33 2 17 < 1 512 . \lVert\overline{\Delta x}'\rVert_1\leq\sqrt{n}\lVert z\rVert_2+n\lVert Q_{H'}w\rVert_\infty\leq8\delta^2n^2(1+\kappa)+n^{3/2}\beta\leq\frac{33}{2^{17}}<\frac{1}{512}. ∥ Δ x ′ ∥ 1 ≤ n ∥ z ∥ 2 + n ∥ Q H ′ w ∥ ∞ ≤ 8 δ 2 n 2 ( 1 + κ ) + n 3/2 β ≤ 2 17 33 < 512 1 . As in the proof of Lemma 6.1 , this ensures x ′ ′ , s ′ ′ > 0 x'',s''>0 x ′′ , s ′′ > 0 q H ′ ′ ≤ 2 q H ′ q_{H''}\leq2q_{H'} q H ′′ ≤ 2 q H ′ ( z + w ) 2 ≤ 2 z 2 + 2 w 2 (z+w)^2\leq2z^2+2w^2 ( z + w ) 2 ≤ 2 z 2 + 2 w 2 (44) :
∥ Q H ′ err ( x ′ ′ , s ′ ′ , μ ′ ) ∥ 2 ≤ 4 ( ∥ Q H ′ z ∥ ∞ ∥ z ∥ 2 + n ∥ Q H ′ w ∥ ∞ 2 ) \lVert Q_{H'}\operatorname{err}(x'',s'',\mu')\rVert_2\leq4\left(\lVert Q_{H'}z\rVert_\infty\lVert z\rVert_2+\sqrt{n}\lVert Q_{H'}w\rVert_\infty^2\right) ∥ Q H ′ err ( x ′′ , s ′′ , μ ′ ) ∥ 2 ≤ 4 ( ∥ Q H ′ z ∥ ∞ ∥ z ∥ 2 + n ∥ Q H ′ w ∥ ∞ 2 ) ≤ 2 14 δ 4 n 3 ( 1 + κ ) 2 + 4 n 3 / 2 β 2 ≤ β 2 14 + β 1024 < β . (60) \begin{aligned}
\leq2^{14}\delta^4n^3(1+\kappa)^2+4n^{3/2}\beta^2 \\
&\leq\frac{\beta}{2^{14}}+\frac{\beta}{1024}<\beta. \tag*{(60)}
\end{aligned} ≤ 2 14 δ 4 n 3 ( 1 + κ ) 2 + 4 n 3/2 β 2 ≤ 2 14 β + 1024 β < β . (60) where in the last inequality we use the intervals of δ \delta δ β \beta β δ ≤ β 1 / 4 128 n 3 / 4 1 + κ \delta\leq\frac{\beta^{1/4}}{128n^{3/4}\sqrt{1+\kappa}} δ ≤ 128 n 3/4 1 + κ β 1/4 ( x ′ ′ , s ′ ′ , μ ′ ) ∈ N R ( β ) (x'',s'',\mu')\in\mathcal{N}^{R}(\beta) ( x ′′ , s ′′ , μ ′ ) ∈ N R ( β )
Deferred proofs from Section 7
In this section, we prove Lemma 7.5, Theorem 7.6, Proposition 7.11 and Proposition 7.12 .
Lemma 7.5. For any M ∈ Q n × n M \in\mathbb{Q}^{n \times n} M ∈ Q n × n R : = C n ( 1 + n ∥ M ∥ max ) n − 1 R := C^{n}(1+n\lVert M\rVert_{\max})^{n-1} R := C n ( 1 + n ∥ M ∥ m a x ) n − 1 C C C M M M ρ 0 S ( M ) < 1 \rho_{0}^{S}(M)<1 ρ 0 S ( M ) < 1 ρ 0 S ( M ) ≤ 1 − 1 R \rho_{0}^{S}(M)\le1-\frac{1}{R} ρ 0 S ( M ) ≤ 1 − R 1 log R = O ( poly ( n , length ( M ) ) ) \log R=O(\operatorname{poly}(n,\operatorname{length}(M))) log R = O ( poly ( n , length ( M )))
Proof. Let S ∈ S n S\in\mathcal{S}^{n} S ∈ S n ( λ j ) j = 1 n (\lambda_j)_{j=1}^{n} ( λ j ) j = 1 n S M SM S M λ i \lambda_i λ i i ∈ [ n ] i\in[n] i ∈ [ n ] S M SM S M ρ 0 S ( M ) < 1 \rho_{0}^{S}(M)<1 ρ 0 S ( M ) < 1 ∣ λ i ∣ < 1 |\lambda_i|<1 ∣ λ i ∣ < 1 det ( ± I n − S M ) ≠ 0 \det(\pm I_n-SM)\ne0 det ( ± I n − S M ) = 0 λ i ≥ 0 \lambda_i\ge0 λ i ≥ 0 det ( I n − S M ) \det(I_n-SM) det ( I n − S M ) 1 / C n 1/C^n 1/ C n ∏ j = 1 n ∣ 1 − λ j ∣ = ∣ det ( I n − S M ) ∣ ≥ 1 / C n \prod_{j=1}^{n}|1-\lambda_j|=|\det(I_n-SM)|\ge1/C^n ∏ j = 1 n ∣1 − λ j ∣ = ∣ det ( I n − S M ) ∣ ≥ 1/ C n j ∈ [ n ] j\in[n] j ∈ [ n ] ∣ λ j ∣ ≤ ∥ S M ∥ ≤ ∥ M ∥ F ≤ n ∥ M ∥ max |\lambda_j|\le\lVert SM\rVert\le\lVert M\rVert_F\le n\lVert M\rVert_{\max} ∣ λ j ∣ ≤ ∥ S M ∥ ≤ ∥ M ∥ F ≤ n ∥ M ∥ m a x
1 − λ i ≥ 1 C n ∏ j ≠ i ∣ 1 − λ j ∣ ≥ 1 C n ∏ j ≠ i ( 1 + ∣ λ j ∣ ) ≥ 1 C n ( 1 + n ∥ M ∥ max ) n − 1 . 1-\lambda_i\ge\frac{1}{C^n\prod_{j\ne i}|1-\lambda_j|}\ge\frac{1}{C^n\prod_{j\ne i}(1+|\lambda_j|)}\ge\frac{1}{C^n(1+n\lVert M\rVert_{\max})^{n-1}}. 1 − λ i ≥ C n ∏ j = i ∣1 − λ j ∣ 1 ≥ C n ∏ j = i ( 1 + ∣ λ j ∣ ) 1 ≥ C n ( 1 + n ∥ M ∥ m a x ) n − 1 1 . giving ∣ λ i ∣ ≤ 1 − 1 C n ( 1 + n ∥ M ∥ max ) n − 1 = 1 − 1 R |\lambda_i|\le1-\frac{1}{C^n(1+n\lVert M\rVert_{\max})^{n-1}}=1-\frac{1}{R} ∣ λ i ∣ ≤ 1 − C n ( 1 + n ∥ M ∥ m a x ) n − 1 1 = 1 − R 1 λ i < 0 \lambda_i<0 λ i < 0 det ( − I n − S M ) \det(-I_n-SM) det ( − I n − S M ) 1 / C n 1/C^n 1/ C n ∏ j = 1 n ∣ 1 + λ j ∣ ≥ 1 / C n \prod_{j=1}^{n}|1+\lambda_j|\ge1/C^n ∏ j = 1 n ∣1 + λ j ∣ ≥ 1/ C n
1 + λ i ≥ 1 C n ∏ j ≠ i ∣ 1 + λ j ∣ ≥ 1 C n ∏ j ≠ i ( 1 + ∣ λ j ∣ ) ≥ 1 C n ( 1 + n ∥ M ∥ max ) n − 1 , 1+\lambda_i\ge\frac{1}{C^n\prod_{j\ne i}|1+\lambda_j|}\ge\frac{1}{C^n\prod_{j\ne i}(1+|\lambda_j|)}\ge\frac{1}{C^n(1+n\lVert M\rVert_{\max})^{n-1}}, 1 + λ i ≥ C n ∏ j = i ∣1 + λ j ∣ 1 ≥ C n ∏ j = i ( 1 + ∣ λ j ∣ ) 1 ≥ C n ( 1 + n ∥ M ∥ m a x ) n − 1 1 , which also gives ∣ λ i ∣ ≤ 1 − 1 C n ( 1 + n ∥ M ∥ max ) n − 1 = 1 − 1 R |\lambda_i|\le1-\frac{1}{C^n(1+n\lVert M\rVert_{\max})^{n-1}}=1-\frac{1}{R} ∣ λ i ∣ ≤ 1 − C n ( 1 + n ∥ M ∥ m a x ) n − 1 1 = 1 − R 1
log R = n log C + ( n − 1 ) log ( 1 + n ∥ M ∥ max ) = O ( poly ( n , length ( M ) ) ) . \log R=n\log C+(n-1)\log(1+n\lVert M\rVert_{\max})=O(\operatorname{poly}(n,\operatorname{length}(M))). log R = n log C + ( n − 1 ) log ( 1 + n ∥ M ∥ m a x ) = O ( poly ( n , length ( M ))) . Theorem 7.6 (Closed Rump’s theorem). For M ∈ R n × n M\in\mathbb{R}^{n\times n} M ∈ R n × n I n − M I_n-M I n − M ρ 0 S ( M ) ≤ 1 \rho_{0}^{S}(M)\le1 ρ 0 S ( M ) ≤ 1 C ( M ) ∈ P 0 C(M)\in\mathcal{P}_0 C ( M ) ∈ P 0
Proof. Suppose that ρ 0 S ( M ) ≤ 1 \rho_{0}^{S}(M)\le1 ρ 0 S ( M ) ≤ 1 ( α k ) k ∈ N = 1 − 1 k + 1 (\alpha_k)_{k\in\mathbb{N}}=1-\frac{1}{k+1} ( α k ) k ∈ N = 1 − k + 1 1 ρ 0 S ( α k M ) = α k ρ 0 S ( M ) ≤ α k < 1 \rho_{0}^{S}(\alpha_kM)=\alpha_k\rho_{0}^{S}(M)\le\alpha_k<1 ρ 0 S ( α k M ) = α k ρ 0 S ( M ) ≤ α k < 1 I n − α k M I_n-\alpha_kM I n − α k M M M M 1 / α k > 1 1/\alpha_k>1 1/ α k > 1 ρ 0 S ( M ) ≤ 1 \rho_{0}^{S}(M)\le1 ρ 0 S ( M ) ≤ 1 Theorem 7.4 , C ( α k M ) = ( I n − α k M ) − 1 ( I n + α k M ) ∈ P C(\alpha_kM)=(I_n-\alpha_kM)^{-1}(I_n+\alpha_kM)\in\mathcal{P} C ( α k M ) = ( I n − α k M ) − 1 ( I n + α k M ) ∈ P lim k → ∞ C ( α k M ) ∈ P 0 \lim_{k\to\infty}C(\alpha_kM)\in\mathcal{P}_0 lim k → ∞ C ( α k M ) ∈ P 0 Lemma 2.4 , where lim k → ∞ C ( α k M ) = lim k → ∞ ( I n − α k M ) − 1 ( I n + α k M ) = ( I n − M ) − 1 ( I n + M ) = C ( M ) \lim_{k\to\infty}C(\alpha_kM)=\lim_{k\to\infty}(I_n-\alpha_kM)^{-1}(I_n+\alpha_kM)=(I_n-M)^{-1}(I_n+M)=C(M) lim k → ∞ C ( α k M ) = lim k → ∞ ( I n − α k M ) − 1 ( I n + α k M ) = ( I n − M ) − 1 ( I n + M ) = C ( M )
Suppose that C ( M ) ∈ P 0 C(M)\in\mathcal{P}_0 C ( M ) ∈ P 0 C ( M ) + ε I n ∈ P C(M)+\varepsilon I_n\in\mathcal{P} C ( M ) + ε I n ∈ P ε > 0 \varepsilon>0 ε > 0 Lemma 2.4 and ( 1 + ε ) I n + C ( M ) (1+\varepsilon)I_n+C(M) ( 1 + ε ) I n + C ( M ) Corollary 2.5 , so N ε : = C − 1 ( C ( M ) + ε I n ) N_\varepsilon:=C^{-1}(C(M)+\varepsilon I_n) N ε := C − 1 ( C ( M ) + ε I n ) Proposition 7.3 and Theorem 7.4 , ρ 0 S ( N ε ) < 1 \rho_{0}^{S}(N_\varepsilon)<1 ρ 0 S ( N ε ) < 1 lim ε → 0 C ( M ) + ε I n = C ( M ) \lim_{\varepsilon\to0}C(M)+\varepsilon I_n=C(M) lim ε → 0 C ( M ) + ε I n = C ( M )
lim ε → 0 N ε = lim ε → 0 ( C ( M ) + ε I n − I n ) ( C ( M ) + ε I n + I n ) − 1 = ( C ( M ) − I n ) ( C ( M ) + I n ) − 1 = C − 1 ( C ( M ) ) = M . \lim_{\varepsilon\to0}N_\varepsilon=\lim_{\varepsilon\to0}(C(M)+\varepsilon I_n-I_n)(C(M)+\varepsilon I_n+I_n)^{-1}=(C(M)-I_n)(C(M)+I_n)^{-1}=C^{-1}(C(M))=M. ε → 0 lim N ε = ε → 0 lim ( C ( M ) + ε I n − I n ) ( C ( M ) + ε I n + I n ) − 1 = ( C ( M ) − I n ) ( C ( M ) + I n ) − 1 = C − 1 ( C ( M )) = M . Therefore, it follows from the continuity of ρ 0 S ( X ) \rho_{0}^{S}(X) ρ 0 S ( X ) X X X [66 Corollary 2.5] that
ρ 0 S ( M ) = ρ 0 S ( lim ε → 0 N ε ) = lim ε → 0 ρ 0 S ( N ε ) ≤ 1. \rho_{0}^{S}(M)=\rho_{0}^{S}\left(\lim_{\varepsilon\to0}N_\varepsilon\right)=\lim_{\varepsilon\to0}\rho_{0}^{S}(N_\varepsilon)\le1. ρ 0 S ( M ) = ρ 0 S ( ε → 0 lim N ε ) = ε → 0 lim ρ 0 S ( N ε ) ≤ 1. Proposition 7.11. Let M ∈ P ⋆ M\in\mathcal{P}_\star M ∈ P ⋆ x ∈ R n x\in\mathbb{R}^{n} x ∈ R n y = 1 2 ( x + M x ) y=\frac{1}{2}(x+Mx) y = 2 1 ( x + M x ) N = ( M − I n ) / 2 N=(M-I_n)/2 N = ( M − I n ) /2
x T M x = ∥ y ∥ 2 − ∥ N ( I n + N ) − 1 y ∥ 2 . x^{\mathsf{T}}Mx=\lVert y\rVert^2-\lVert N(I_n+N)^{-1}y\rVert^2. x T M x = ∥ y ∥ 2 − ∥ N ( I n + N ) − 1 y ∥ 2 . If N 2 = 0 n × n N^2=0_{n\times n} N 2 = 0 n × n x T M x = y T ( I n − N T N ) y x^{\mathsf{T}}Mx=y^{\mathsf{T}}(I_n-N^{\mathsf{T}}N)y x T M x = y T ( I n − N T N ) y
Proof. By Lemma 2.4 , I n + M I_n+M I n + M y = 1 2 ( I n + M ) x y=\frac{1}{2}(I_n+M)x y = 2 1 ( I n + M ) x
M x = 2 M ( I n + M ) − 1 y = 2 ( I n − ( I n + M ) − 1 ) y = 2 ( I n − 1 2 ( I n + N ) − 1 ) y = ( I n + N ( I n + N ) − 1 ) y , Mx=2M(I_n+M)^{-1}y=2\left(I_n-(I_n+M)^{-1}\right)y=2\left(I_n-\frac{1}{2}(I_n+N)^{-1}\right)y=(I_n+N(I_n+N)^{-1})y, M x = 2 M ( I n + M ) − 1 y = 2 ( I n − ( I n + M ) − 1 ) y = 2 ( I n − 2 1 ( I n + N ) − 1 ) y = ( I n + N ( I n + N ) − 1 ) y , and x = 2 ( I n + M ) − 1 y = ( I n + N ) − 1 y = ( I n − N ( I n + N ) − 1 ) y x=2(I_n+M)^{-1}y=(I_n+N)^{-1}y=(I_n-N(I_n+N)^{-1})y x = 2 ( I n + M ) − 1 y = ( I n + N ) − 1 y = ( I n − N ( I n + N ) − 1 ) y y T ( I n − A ) T ( I n + A ) y = ∥ y ∥ 2 − ∥ A y ∥ 2 y^{\mathsf{T}}(I_n-A)^{\mathsf{T}}(I_n+A)y=\lVert y\rVert^2-\lVert Ay\rVert^2 y T ( I n − A ) T ( I n + A ) y = ∥ y ∥ 2 − ∥ A y ∥ 2 A ∈ R n × n A\in\mathbb{R}^{n\times n} A ∈ R n × n
x T M x = y T ( I n − N ( I n + N ) − 1 ) T ( I n + N ( I n + N ) − 1 ) y = ∥ y ∥ 2 − ∥ N ( I n + N ) − 1 y ∥ 2 x^{\mathsf{T}}Mx=y^{\mathsf{T}}\left(I_n-N(I_n+N)^{-1}\right)^{\mathsf{T}}\left(I_n+N(I_n+N)^{-1}\right)y=\lVert y\rVert^2-\lVert N(I_n+N)^{-1}y\rVert^2 x T M x = y T ( I n − N ( I n + N ) − 1 ) T ( I n + N ( I n + N ) − 1 ) y = ∥ y ∥ 2 − ∥ N ( I n + N ) − 1 y ∥ 2 When N N N N 2 = 0 n × n N^2 = 0_{n \times n} N 2 = 0 n × n ( I n + N ) ( I n − N ) = I n (I_n + N)(I_n - N) = I_n ( I n + N ) ( I n − N ) = I n ( I n + N ) − 1 = I n − N (I_n + N)^{-1} = I_n - N ( I n + N ) − 1 = I n − N
x ⊤ M x = ∥ y ∥ 2 − ∥ N ( I n − N ) y ∥ 2 = ∥ y ∥ 2 − ∥ N y ∥ 2 = y ⊤ ( I n − N ⊤ N ) y . x^\top Mx = \|y\|^2 - \|N(I_n - N)y\|^2 = \|y\|^2 - \|Ny\|^2 = y^\top(I_n - N^\top N)y . x ⊤ M x = ∥ y ∥ 2 − ∥ N ( I n − N ) y ∥ 2 = ∥ y ∥ 2 − ∥ N y ∥ 2 = y ⊤ ( I n − N ⊤ N ) y . Proposition 7.12. Let G = ( V , E ) G = (V,E) G = ( V , E ) ∣ V ∣ = n ≥ 2 |V| = n \ge2 ∣ V ∣ = n ≥ 2 ∣ E ∣ = m |E| = m ∣ E ∣ = m v ∈ R n v \in\mathbb{R}^n v ∈ R n
v ⊤ ( L G − I n ) v ≤ ∥ v ∥ ∞ 2 ( 4 maxcut ( G ) − n ) . v^\top(L_G - I_n)v \le\|v\|_\infty^2(4\operatorname{maxcut}(G) - n). v ⊤ ( L G − I n ) v ≤ ∥ v ∥ ∞ 2 ( 4 maxcut ( G ) − n ) . Proof. If v = 0 v = 0 v = 0 v ≠ 0 v \ne0 v = 0 v ^ = v / ∥ v ∥ ∞ \hat{v} = v/\|v\|_\infty v ^ = v /∥ v ∥ ∞
v ⊤ ( L G − I n ) v = ∥ v ∥ ∞ 2 ⋅ v ^ ⊤ ( L G − I n ) v ^ ≤ ∥ v ∥ ∞ 2 ⋅ max s ∈ [ − 1 , 1 ] n s ⊤ ( L G − I n ) s . v^\top(L_G - I_n)v = \|v\|_\infty^2 \cdot\hat{v}^\top(L_G - I_n)\hat{v} \le\|v\|_\infty^2 \cdot\max_{s \in[-1,1]^n} s^\top(L_G - I_n)s . v ⊤ ( L G − I n ) v = ∥ v ∥ ∞ 2 ⋅ v ^ ⊤ ( L G − I n ) v ^ ≤ ∥ v ∥ ∞ 2 ⋅ s ∈ [ − 1 , 1 ] n max s ⊤ ( L G − I n ) s . The maximum of s ∈ [ − 1 , 1 ] n s \in[-1,1]^n s ∈ [ − 1 , 1 ] n s ∈ { − 1 , 1 } n s \in\{-1,1\}^n s ∈ { − 1 , 1 } n [65 Corollary 8.1] . To see the proof of this, note that ( L G − I n ) i i ≥ 0 (L_G - I_n)_{ii} \ge0 ( L G − I n ) ii ≥ 0 i ∈ [ n ] i \in[n] i ∈ [ n ] G G G ≥ 1 \ge1 ≥ 1 f ( s ) = s ⊤ ( L G − I n ) s f(s) = s^\top(L_G - I_n)s f ( s ) = s ⊤ ( L G − I n ) s s i s_i s i s ∈ { − 1 , 1 } n s \in\{-1,1\}^n s ∈ { − 1 , 1 } n
v ⊤ ( L G − I n ) v ≤ ∥ v ∥ ∞ 2 ⋅ max s ∈ { − 1 , 1 } n s ⊤ ( L G − I n ) s = ∥ v ∥ ∞ 2 ( 4 maxcut ( G ) − n ) , v^\top(L_G - I_n)v \le\|v\|_\infty^2 \cdot\max_{s \in\{-1,1\}^n} s^\top(L_G - I_n)s = \|v\|_\infty^2 (4\operatorname{maxcut}(G) - n) , v ⊤ ( L G − I n ) v ≤ ∥ v ∥ ∞ 2 ⋅ s ∈ { − 1 , 1 } n max s ⊤ ( L G − I n ) s = ∥ v ∥ ∞ 2 ( 4 maxcut ( G ) − n ) , where the last equality follows by Proposition 7.10.
Corollary 7.18. Let M ∈ P ⋆ ∩ R n × n M \in\mathcal{P}_\star\cap\mathbb{R}^{n \times n} M ∈ P ⋆ ∩ R n × n M M M ker ( M p ) = ker ( M ) \ker(M^p) = \ker(M) ker ( M p ) = ker ( M ) rk ( M p ) = rk ( M ) \operatorname{rk}(M^p) = \operatorname{rk}(M) rk ( M p ) = rk ( M ) p ∈ N p \in\mathbb{N} p ∈ N sgn ( ker ( M ) ) = sgn ( ker ( M ⊤ ) ) \operatorname{sgn}(\ker(M)) = \operatorname{sgn}(\ker(M^\top)) sgn ( ker ( M )) = sgn ( ker ( M ⊤ ))
Proof. Suppose 0 is not a semi-simple eigenvalue of M M M x , y ∈ R n x,y \in\mathbb{R}^n x , y ∈ R n x ≠ 0 x \ne0 x = 0 M x = 0 Mx = 0 M x = 0 M y = x My = x M y = x M ∈ P ⋆ M \in\mathcal{P}_\star M ∈ P ⋆ I n + M I_n + M I n + M Corollary 2.5 . Let M ~ = C − 1 ( M ) = ( M − I ) ( M + I ) − 1 \widetilde{M} = C^{-1}(M) = (M - I)(M + I)^{-1} M = C − 1 ( M ) = ( M − I ) ( M + I ) − 1 M ~ x = − x \widetilde{M}x = -x M x = − x M ~ y = ( M − I ) ( y − ( I + M ) − 1 x ) = ( M − I ) ( y − x ) = 2 x − y \widetilde{M}y = (M - I)(y - (I + M)^{-1}x) = (M - I)(y - x) = 2x - y M y = ( M − I ) ( y − ( I + M ) − 1 x ) = ( M − I ) ( y − x ) = 2 x − y ∣ M ~ x ∣ = ∣ x ∣ |\widetilde{M}x| = |x| ∣ M x ∣ = ∣ x ∣
M ~ x ∘ M ~ ( − y ) − x ∘ ( − y ) = ( − x ) ∘ ( y − 2 x ) + x ∘ y = 2 x 2 ≥ 0 , \widetilde{M}x \circ\widetilde{M}(-y) - x \circ(-y) = (-x) \circ(y - 2x) + x \circ y = 2x^2 \ge0 , M x ∘ M ( − y ) − x ∘ ( − y ) = ( − x ) ∘ ( y − 2 x ) + x ∘ y = 2 x 2 ≥ 0 , Theorem 7.17 gives M ~ x ∘ M ~ y = x ∘ y \widetilde{M}x \circ\widetilde{M}y = x \circ y M x ∘ M y = x ∘ y x = 0 x = 0 x = 0 M p M^p M p M M M p ∈ N p \in\mathbb{N} p ∈ N
Let x ∈ ker ( M ) x \in\ker(M) x ∈ ker ( M ) M ~ x = − x \widetilde{M}x = -x M x = − x Theorem 7.17 ensures that if ( − x ) ∘ M ~ y ≥ x ∘ y (-x) \circ\widetilde{M}y \ge x \circ y ( − x ) ∘ M y ≥ x ∘ y ( − x ) ∘ M ~ y = x ∘ y (-x) \circ\widetilde{M}y = x \circ y ( − x ) ∘ M y = x ∘ y y ∈ R n y \in\mathbb{R}^n y ∈ R n A x : = − diag ( x ) ( I + M ~ ) A_x := -\operatorname{diag}(x)(I + \widetilde{M}) A x := − diag ( x ) ( I + M ) y ∈ R n y \in\mathbb{R}^n y ∈ R n A x y ≥ 0 A_xy \ge0 A x y ≥ 0 A x y = 0 A_xy = 0 A x y = 0 y ∈ R n y \in\mathbb{R}^n y ∈ R n A x y ≥ 0 A_xy \ge0 A x y ≥ 0 A x y ≠ 0 A_xy \ne0 A x y = 0 w > 0 w > 0 w > 0 A x ⊤ w = 0 A_x^\top w = 0 A x ⊤ w = 0 x ~ : = diag ( w ) x \widetilde{x} := \operatorname{diag}(w)x x := diag ( w ) x ( I + M ~ ⊤ ) x ~ = ( I + M ~ ⊤ ) diag ( x ) w = − A x ⊤ w = 0 (I + \widetilde{M}^\top)\widetilde{x} = (I + \widetilde{M}^\top)\operatorname{diag}(x)w = -A_x^\top w = 0 ( I + M ⊤ ) x = ( I + M ⊤ ) diag ( x ) w = − A x ⊤ w = 0 x ~ ∈ ker ( I + M ~ ⊤ ) \widetilde{x} \in\ker(I + \widetilde{M}^\top) x ∈ ker ( I + M ⊤ )
I + M ~ ⊤ = I + ( M ⊤ + I ) − 1 ( M ⊤ − I ) = 2 ( M ⊤ + I ) − 1 M ⊤ , I + \widetilde{M}^\top= I + (M^\top+ I)^{-1}(M^\top- I) = 2(M^\top+ I)^{-1}M^\top, I + M ⊤ = I + ( M ⊤ + I ) − 1 ( M ⊤ − I ) = 2 ( M ⊤ + I ) − 1 M ⊤ , ker ( I + M ~ ⊤ ) = ker ( M ⊤ ) \ker(I + \widetilde{M}^\top) = \ker(M^\top) ker ( I + M ⊤ ) = ker ( M ⊤ ) x ~ ∈ ker ( M ⊤ ) \widetilde{x} \in\ker(M^\top) x ∈ ker ( M ⊤ ) sgn ( x ~ ) = sgn ( x ) \operatorname{sgn}(\widetilde{x}) = \operatorname{sgn}(x) sgn ( x ) = sgn ( x ) x ~ = diag ( w ) x \widetilde{x} = \operatorname{diag}(w)x x = diag ( w ) x w > 0 w > 0 w > 0 sgn ( ker ( M ) ) ⊆ sgn ( ker ( M ⊤ ) ) \operatorname{sgn}(\ker(M)) \subseteq\operatorname{sgn}(\ker(M^\top)) sgn ( ker ( M )) ⊆ sgn ( ker ( M ⊤ )) sgn ( ker ( M ) ) = sgn ( ker ( M ⊤ ) ) \operatorname{sgn}(\ker(M)) = \operatorname{sgn}(\ker(M^\top)) sgn ( ker ( M )) = sgn ( ker ( M ⊤ )) M ⊤ M^\top M ⊤ sgn ( ker ( M ⊤ ) ) ⊆ sgn ( ker ( M ) ) \operatorname{sgn}(\ker(M^\top)) \subseteq\operatorname{sgn}(\ker(M)) sgn ( ker ( M ⊤ )) ⊆ sgn ( ker ( M )) M ⊤ ∈ P ⋆ M^\top\in\mathcal{P}_\star M ⊤ ∈ P ⋆ Theorem 2.6 .
AI Disclosure
The authors uploaded to ChatGPT a writeup of their own simple IPM analysis from Section 4 and characterization from Theorem 1.1, seeking to avoid the linear dependence on the handicap number present in this analysis and in earlier work. Prompted to test a range of approaches, ChatGPT suggested measuring the Newton step H e r r Herr H er r M M M ( I − ∣ H ∣ / α ) − 1 (I - |H|/\alpha)^{-1} ( I − ∣ H ∣/ α ) − 1 α \alpha α H H H § 5.4 and § 5.5 .
DeepSeek assisted with the hardness results, and independently suggested using Rump’s sign spectral radius to prove coNP-hardness of testing sufficiency. The authors refined all ideas and strengthened the arguments.
We view the connection to Rump’s work, which appears to have been overlooked in the IPM literature, as the most significant LLM contribution. LLM use in writing was limited to proofreading and helping to improve the flow of the text.
References 
