Relaxing Nonconvex Quadratic Functions by Multiple Adaptive Diagonal Perturbations

Abstract The current bottleneck of globally solving mixed-integer (nonconvex) quadratically constrained problems (MIQCPs) is still to construct strong but computationally cheap convex relaxations, especially when dense quadratic functions are present. We propose a cutting-surface method based on multiple diagonal perturbations to derive convex quadratic relaxations for nonconvex quadratic problems with separable constraints. Our relaxations can also be realized as outer-approximations of a semi-infinite convex formulation of a semidefinite relaxation proposed by Buchheim and Wiegele. The corresponding separation problem is a highly structured semidefinite program (SDP) with a convex but nonsmooth objective function. We propose to solve this separation problem with a specialized barrier coordinate minimization algorithm. Numerical experiments on randomly generated instances show that our approach is very promising. We also discuss how to apply our method to more general cases of MIQCPs.