Second Order Cone Constrained Convex Relaxations for Nonconvex Quadratically Constrained Quadratic Programming

TL;DR

This paper introduces new convex relaxation techniques for nonconvex QCQP problems using second order cone constraints, extending RLT-like methods to improve relaxation quality and reduce gaps.

Contribution

It develops SOC-based relaxations for nonconvex QCQP, extending RLT-like techniques to various constraint pairs and demonstrating effectiveness through numerical experiments.

Findings

SOC relaxations improve solution bounds for QCQP

RLT-like techniques reduce relaxation gaps

Numerical results show enhanced relaxation performance

Abstract

In this paper, we present new convex relaxations for nonconvex quadratically constrained quadratic programming (QCQP) problems. While recent research has focused on strengthening convex relaxations using reformulation-linearization technique (RLT), the state-of-the-art methods lose their effectiveness when dealing with (multiple) nonconvex quadratic constraints in QCQP. In this research, we decompose and relax each nonconvex constraint to two second order cone (SOC) constraints and then linearize the products of the SOC constraints and linear constraints to construct some effective new valid constraints. Moreover, we extend the reach of the RLT-like techniques for almost all different types of constraint-pairs (including valid inequalities by linearizing the product of a pair of SOC constraints, and the the Hadamard product or the Kronecker product of two respective valid linear matrix inequalities), examine dominance relationships among different valid inequalities, and explore almost all possibilities of gaining benefits from generating valid constraints. Especially, we successfully demonstrate that applying RLT-like techniques to additional redundant linear constraints could reduce the relaxation gap. We demonstrate the efficiency of our results with numerical experiments.