Spectral Relaxations and Branching Strategies for Global Optimization of Mixed-Integer Quadratic Programs

Abstract

We consider the global optimization of nonconvex (mixed-integer) quadratic programs. We present a family of convex quadratic relaxations derived by convexifying nonconvex quadratic functions through perturbations of the quadratic matrix. We investigate the theoretical properties of these relaxations and show that they are equivalent to some particular semidefinite programs. We also introduce novel branching variable selection strategies motivated by the quadratic relaxations investigated in this paper. The proposed relaxation and branching techniques are implemented in the global optimization solver BARON and tested by conducting numerical experiments on a large collection of problems. Results demonstrate that the proposed implementation leads to very significant reductions in BARON's computational times to solve the test problems.