Exploiting Sparsity in Semidefinite Programming via Matrix Completion I: General Framework

Abstract

A critical disadvantage of primal-dual interior-point methods compared to dual interior-point methods for large scale semidefinite programs (SDPs) has been that the primal positive semidefinite matrix variable becomes fully dense in general even when all data matrices are sparse. Based on some fundamental results about positive semidefinite matrix completion, this article proposes a general method of exploiting the aggregate sparsity pattern over all data matrices to overcome this disadvantage. Our method is used in two ways. One is a conversion of a sparse SDP having a large scale positive semidefinite matrix variable into an SDP having multiple but smaller positive semidefinite matrix variables to which we can effectively apply any interior-point method for SDPs employing a standard block-diagonal matrix data structure. The other way is an incorporation of our method into primal-dual interior-point methods which we can apply directly to a given SDP. In Part II of this article, we will investigate an implementation of such a primal-dual interior-point method based on positive definite matrix completion, and report some numerical results.

Conic Optimization Software

“…Efficient parallel versions [35,36] are released for both shared and distributed memory environments. SDPA also implements techniques to decompose sparse problems using matrix completion [37][38][39] methods. A Matlab interface [40] is also available.…”

“…These methods [34,[36][37][38][39] exploit the fact that the dual slack variable S inherits the sparsity structure of the coefficient matrices. By enforcing positive semidefiniteness only on a small number of carefully chosen submatrices of a large sparse matrix, one can make sure that the original matrix can be completed to be positive semidefinite.…”

A feasible direction interior point algorithm for nonlinear semidefinite programming

“…Applications in a wide range of areas lead to semidefinite programming problems. For example, combinatorial optimization [3], nonconvex quadratic programming [16], eigenvalue optimization [25], systems control theory [12], matrix completion problems [28], problems in statistics [15] and structural design [2]. In particular SDP was considered to solve free material problems in structural mechanical design, see for example [37], [24], [35], [34] and [38].…”

Faster, but weaker, relaxations for quadratically constrained quadratic programs

“…Semidefinite programming (SDP) relaxations can provide tight bounds [6,11,12], but they can also be expensive to solve by classical interior-point methods [14]. Many researchers have thus suggested various alternatives to interior-point methods for improving solution times [1,2,4,8,15,19]. These efforts have been quite successful on many classes of SDP relaxations.…”