2x2 convexifications for convex quadratic optimization with indicator variables

TL;DR

This paper develops new convexification techniques for quadratic optimization with indicator variables, providing stronger relaxations and formulations that improve solution quality and reduce integrality gaps.

Contribution

It introduces a novel 2x2 convexification approach for the bivariate case and extends it to a stronger SDP relaxation for the general problem.

Findings

The new SDP relaxation outperforms existing relaxations in reducing the integrality gap.

Convex hull descriptions improve the tightness of the relaxations.

Computational results demonstrate the effectiveness of the proposed formulations.

Abstract

In this paper, we study the convex quadratic optimization problem with indicator variables. For the bivariate case, we describe the convex hull of the epigraph in the original space of variables, and also give a conic quadratic extended formulation. Then, using the convex hull description for the bivariate case as a building block, we derive an extended SDP relaxation for the general case. This new formulation is stronger than other SDP relaxations proposed in the literature for the problem, including Shor's SDP relaxation, the optimal perspective relaxation as well as the optimal rank-one relaxation. Computational experiments indicate that the proposed formulations are quite effective in reducing the integrality gap of the optimization problems.