A low-dimensional SDP relaxation based spatial branch and bound method for nonconvex quadratic programs

Abstract

In this paper, we propose a novel low-dimensional semidefinite programming (SDP) relaxation for convex quadratically constrained nonconvex quadratic programming problems. This new relaxation is derived by simultaneous matrix diagonalization method under the difference of convex decomposition scheme. The highlight of the relaxation is the low dimensionality of the positive semidefinite constraint, which only depends on the number of negative eigenvalues in the objective function. We prove that a mixed SOCP and SDP relaxation appeared in the literature is equivalent to the proposed relaxation, while the proposed relaxation has fewer constraints. We also provide conditions under which the proposed relaxation is as tight as the classical SDP relaxation and provides an optimal value for the original problem. For general cases, a spatial branch-and-bound algorithm is designed for finding a global optimal solution. Extensive numerical experiments support that the proposed algorithm outperforms two cutting-edge algorithms when the problem size is medium or large and the number of negative eigenvalues in the nonconvex objective function is relatively small.