Exact SDP relaxations for quadratic programs with bipartite graph structures

TL;DR

This paper proves that SDP relaxations are exact for certain nonconvex quadratic programs with bipartite graph structures, and introduces a conversion method to extend this to broader classes of problems.

Contribution

The paper establishes exactness conditions for SDP relaxations in bipartite structured QCQPs and proposes a conversion method to handle more general QCQPs.

Findings

SDP relaxation is exact for bipartite structured QCQPs under certain conditions.

A conversion method broadens the class of QCQPs solvable by SDP relaxation.

Numerical examples demonstrate the effectiveness of the approach.

Abstract

For nonconvex quadratically constrained quadratic programs (QCQPs), we first show that, under certain feasibility conditions, the standard semidefinite (SDP) relaxation is exact for QCQPs with bipartite graph structures. The exact optimal solutions are obtained by examining the dual SDP relaxation and the rank of the optimal solution of this dual SDP relaxation under strong duality. Our results on the QCQPs generalize the results on QCQP with sign-definite bipartite graph structures, QCQPs with forest structures, and QCQPs with nonpositive off-diagonal data elements. Second, we propose a conversion method from QCQPs with no particular structure to the ones with bipartite graph structures. As a result, we demonstrate that a wider class of QCQPs can be exactly solved by the SDP relaxation. Numerical instances are presented for illustration.