Finding Sparse Solutions for Packing and Covering Semidefinite Programs

TL;DR

This paper introduces an algorithm for finding sparse dual solutions in packing and covering semidefinite programs, improving efficiency and support size independence, based on an extension of a logarithmic-potential method originally for linear programs.

Contribution

It extends a logarithmic-potential algorithm to semidefinite programs, enabling the computation of sparse dual solutions with support size independent of the number of constraints.

Findings

Algorithm successfully finds sparse dual solutions.

Support size of solutions is nearly independent of primal constraints.

Method improves efficiency over general SDP solvers.

Abstract

Packing and covering semidefinite programs (SDPs) appear in natural relaxations of many combinatorial optimization problems as well as a number of other applications. Recently, several techniques were proposed, that utilize the particular structure of this class of problems, to obtain more efficient algorithms than those offered by general SDP solvers. For certain applications, such as those described in this paper, it may be desirable to obtain {\it sparse} dual solutions, i.e., those with support size (almost) independent of the number of primal constraints. In this paper, we give an algorithm that finds such solutions, which is an extension of a {\it logarithmic-potential} based algorithm of Grigoriadis, Khachiyan, Porkolab and Villavicencio (SIAM Journal of Optimization 41 (2001)) for packing/covering linear programs.