Exact SDP relaxations for quadratic programs with bipartite graph structures

Abstract

For nonconvex quadratically constrained quadratic programs (QCQPs), we first show that, under certain feasibility conditions, the standard semidefinite programming (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 generalize the previous results on QCQPs 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. © 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.