A second-order cone representable class of nonconvex quadratic programs

TL;DR

This paper introduces a novel second-order cone relaxation for sparse nonconvex quadratic programs, extending RLT and identifying conditions for polynomial-size formulations that match the original problem's optimal value.

Contribution

It develops an SOC-representable relaxation for sparse nonconvex quadratic programs and finds structural conditions for polynomial-size formulations with exact solutions.

Findings

Proposed a second-order cone relaxation for sparse nonconvex quadratic programs.

Identified structural conditions for polynomial-size SOC formulations.

Established when the relaxation's optimal value equals the original problem's value.

Abstract

We consider the problem of minimizing a sparse nonconvex quadratic function over the unit hypercube. By developing an extension of the Reformulation-Linearization Technique (RLT) to continuous quadratic sets, we propose a novel second-order cone (SOC) representable relaxation for this problem. By exploiting the sparsity of the quadratic function, we establish a sufficient condition under which the convex hull of the feasible region of the lifted quadratic program is SOC-representable. While the proposed formulation may be of exponential size in general, we identify additional structural conditions that guarantee the existence of a polynomial-size SOC-representable formulation, which can be constructed in polynomial time. Under these conditions, the optimal value of the nonconvex quadratic program coincides with that of a polynomial-size second-order cone program. Our results serve as a starting point for bridging the gap between the Boolean quadric polytope of sparse problems and its continuous counterpart.