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

TLDR

Primal‑dual interior‑point methods for large‑scale semidefinite programs suffer from dense primal matrix variables even when all data matrices are sparse. This paper proposes a general approach that exploits the aggregate sparsity pattern across all data matrices to alleviate that density problem. The method either decomposes a large sparse SDP into multiple smaller positive‑semidefinite blocks or incorporates the sparsity exploitation directly into primal‑dual interior‑point algorithms, enabling efficient block‑diagonal processing.

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.