On Tractable Convex Relaxations of Standard Quadratic Optimization Problems under Sparsity Constraints

TL;DR

This paper investigates convex relaxations for sparse standard quadratic optimization problems, analyzing their structural properties, solution quality, and conditions for exactness to improve tractability in applications.

Contribution

It introduces and compares convex relaxations for sparse StQPs, revealing structural insights and conditions for their exactness, advancing the understanding of their solution bounds.

Findings

Structural properties of relaxations related to sparsity constraints

Conditions ensuring the exactness of relaxations

Bounds on the quality of solutions from different relaxations

Abstract

Standard quadratic optimization problems (StQPs) provide a versatile modelling tool in various applications. In this paper, we consider StQPs with a hard sparsity constraint, referred to as sparse StQPs. We focus on various tractable convex relaxations of sparse StQPs arising from a mixed-binary quadratic formulation, namely, the linear optimization relaxation given by the reformulation-linearization technique, the Shor relaxation, and the relaxation resulting from their combination. We establish several structural properties of these relaxations in relation to the corresponding relaxations of StQPs without any sparsity constraints, and pay particular attention to the rank-one feasible solutions retained by these relaxations. We then utilize these relations to establish several results about the quality of the lower bounds arising from different relaxations. We also present several conditions that ensure the exactness of each relaxation.