Sparse Cuts for the Positive Semidefinite Cone

TL;DR

This paper introduces sparse linear inequalities that approximate the positive semidefinite cone, enabling LP relaxations to match SDP bounds and improve computational efficiency in nonconvex optimization.

Contribution

It presents a method to identify sparse inequalities supporting LP relaxations that are as tight as SDP relaxations for nonconvex quadratic problems.

Findings

Sparse LP relaxations match SDP bounds

The approach accelerates branch-and-bound algorithms

Structured projection SDP aids in identifying inequalities

Abstract

We consider optimization problems containing nonconvex quadratic functions for which semidefinite programming (SDP) relaxations often yield strong bounds. We investigate linear inequalities that outer approximate the positive semidefinite cone and are sparse in the sense that they are supported only on the variables corresponding to products of variables present in quadratic functions. We show that these sparse linear inequalities yield an LP relaxation that gives the same bound as the SDP relaxation. We demonstrate how to identify these inequalities via a separation procedure that involves solving a structured ``projection'' SDP. In a computational study, we find that the sparse LP relaxations defined by these inequalities can accelerate branch-and-bound methods for globally solving nonconvex optimization problems.