Polyhedral approximations of the semidefinite cone and their application

Abstract

We develop techniques to construct a series of sparse polyhedral approximations of the semidefinite cone. Motivated by the semidefinite (SD) bases proposed by Tanaka and Yoshise (Ann Oper Res 265:155–182, 2018), we propose a simple expansion of SD bases so as to keep the sparsity of the matrices composing it. We prove that the polyhedral approximation using our expanded SD bases contains the set of all diagonally dominant matrices and is contained in the set of all scaled diagonally dominant matrices. We also prove that the set of all scaled diagonally dominant matrices can be expressed using an infinite number of expanded SD bases. We use our approximations as the initial approximation in cutting plane methods for solving a semidefinite relaxation of the maximum stable set problem. It is found that the proposed methods with expanded SD bases are significantly more efficient than methods using other existing approximations or solving semidefinite relaxation problems directly.

Evaluating approximations of the semidefinite cone with trace normalized distance

“…In [14], the authors proposed a class of polyhedral approximations of the semidefinite cone, denoted as SDB * n . The experimental results on cutting-plane methods, where SDB * n is used as the approximation of S n + , for solving SDPs are promising.…”

“…These duality relationships imply that SDD * n is a subset of DD * n . It is worth noting that DD * n and SDD * n are used as approximations of the semidefinite cone in cutting-plane methods ([1], [2], [4], [14]) and facial reduction methods [11].…”

“…Several sets have been used as approximations of the semidefinite cone, including the k-PSD closure, namely S n,k ( [5]) and the dual cone of the set of diagonally dominant matrices (resp., scaled diagonally dominant matrices), namely DD * n (resp., SDD * n ) ( [2]). Multiple experimental results have shown the efficiency of cutting-plane methods using these approximations ( [2], [4], [14]). Although the inclusive relationship of these approximations has been given (see, e.g.…”

Iterative Inner/outer Approximations for Scalable Semidefinite Programs using Block Factor-width-two Matrices

“…While the SDD approximation brings considerable computational efficiency in solving (1), the solution might be very conservative [5], [14]. Several iterative methods have been further proposed to improve solution quality, such as adding linear cuts or second-order cuts [15]- [17], and basis pursuit searching [18], [19]. These methods [15]- [19] solves a linear program (LP) or an SOCP at each iteration, but may require many iterations to get a reasonable good solution (if possible) .…”

Chordal and factor-width decompositions for scalable semidefinite and polynomial optimization

“…At the same time, attempting an approximation over FW n 3 will result into an O(n 3 ) number of positive semidefinite constraints, which may not strike a good balance between approximation and computational efficiency. For this reason, most work has focused on the case of factor-width-two matrices and on some closely related extensions (Ahmadi et al, 2017a;Ahmadi & Hall, 2017;Wang et al, 2021c).…”