Polyhedral approximations of the semidefinite cone and their application

Wang, Yuzhu; Tanaka, Akihiro; Yoshise, Akiko

doi:10.48550/arxiv.1905.00166

Cited by 2 publications

(2 citation statements)

References 31 publications

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“…It is worth noting that the dual cone of S n,k is the set of symmetric matrices with factor width k, defined and studied in [7]. In particular, the set of symmetric matrices with factor width 2 is the set of scaled diagonally dominant matrices [29], i.e., symmetric matrices A such that DAD is diagonally dominant for some positive diagonal matrix D. Note that [1] uses scaled diagonally dominant for constructing inner approximation of the SDP cones for use in solving polynomial optimization problems.…”

Section: Introduction 1motivationmentioning

confidence: 99%

Sparse PSD approximation of the PSD cone

Blekherman

Dey

Molinaro

et al. 2020

Preprint

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While semidefinite programming (SDP) problems are polynomially solvable in theory, it is often difficult to solve large SDP instances in practice. One technique to address this issue is to relax the global positive-semidefiniteness (PSD) constraint and only enforce PSD-ness on smaller k × k principal submatrices -we call this the sparse SDP relaxation. Surprisingly, it has been observed empirically that in some cases this approach appears to produce bounds that are close to the optimal objective function value of the original SDP. In this paper, we formally attempt to compare the strength of the sparse SDP relaxation vis-à-vis the original SDP from a theoretical perspective.In order to simplify the question, we arrive at a data independent version of it, where we compare the sizes of SDP cone and the k-PSD closure, which is the cone of matrices where PSDness is enforced on all k × k principal submatrices. In particular, we investigate the question of how far a matrix of unit Frobenius norm in the k-PSD closure can be from the SDP cone. We provide two incomparable upper bounds on this farthest distance as a function of k and n. We also provide matching lower bounds, which show that the upper bounds are tight within a constant in different regimes of k and n. Other than linear algebra techniques, we extensively use probabilistic methods to arrive at these bounds. One of the lower bounds is obtained by observing a connection between matrices in the k-PSD closure and matrices satisfying the restricted isometry property (RIP).

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Section: Introduction 1motivationmentioning

confidence: 99%

Sparse PSD approximation of the PSD cone

Blekherman

Dey

Molinaro

et al. 2020

Preprint

View full text Add to dashboard Cite

show abstract

“…The set S n,k is a closed convex cone and its dual cone is the set of symmetric matrices with factor width k, defined and studied in [5,18,9]. The set of symmetric matrices with factor width 2 is the set of scaled diagonally dominant matrices [5,24], i.e., symmetric matrices A such that DAD is diagonally dominant for some positive diagonal matrix D. The paper [1] uses scaled diagonally dominant matrices for constructing inner approximation of the SDP cones for use in solving polynomial optimization problems. See [13,21,22] for related papers.…”

mentioning

confidence: 99%

Hyperbolic Relaxation of $k$-Locally Positive Semidefinite Matrices

Blekherman¹,

Dey²,

Shu³

et al. 2020

Preprint

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In order to solve positive semidefinite (PSD) programs efficiently, a successful computational trick is to consider a relaxation, where PSD-ness is enforced only on a collection of submatrices. In order to study this formally, we consider the class of n × n symmetric matrices where we enforce PSD-ness on all k × k principal submatrices. We call a matrix in this class k-locally PSD. In order to compare the set of k-locally PSD matrices (denoted as S n,k ) to the set of PSD matrices, we study eigenvalues of k-locally PSD matrices. The key insight in this paper is that the eigenvalues of a matrix in S n,k are contained in a convex set, H(e n k ) (this can be defined as the hyperbolicity cone of the elementary symmetric polynomial e k n (where e n k (x) = S⊆[n]:|S|=k i∈S x i ) with respect to the all ones vector). Using this insight, we are able to improve previously known upper bounds on the Frobenius distance between matrices in S n,k from PSD matrices. We also study the quality of the convex relaxation H(e n k ). We first show that this relaxation is tight for the case of k = n − 1, that is, for every vector in H(e n n−1 ) there exists a matrix in S n,n−1 whose eigenvalues are equal to the components of the vector. We then prove a structure theorem that precisely characterizes the non-singular matrices in S n,k whose eigenvalues belong to the boundary of H(e n k ). This result indicates that "large parts" of the boundary of H(e n k ) do not intersect with the eigenvalues of matrices in S n,k when k < n − 1.

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Polyhedral approximations of the semidefinite cone and their application

Cited by 2 publications

References 31 publications

Sparse PSD approximation of the PSD cone

Sparse PSD approximation of the PSD cone

Hyperbolic Relaxation of $k$-Locally Positive Semidefinite Matrices

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