SDP-quality bounds via convex quadratic relaxations for global optimization of mixed-integer quadratic programs

TL;DR

This paper introduces a new class of convex quadratic relaxations for nonconvex mixed-integer quadratic programs, improving global optimization performance by leveraging quadratic cuts and specialized solution algorithms.

Contribution

The paper develops a novel convex quadratic relaxation method using quadratic cuts derived from a structured linear matrix inequality, enhancing existing semidefinite relaxations.

Findings

Significant performance improvements in BARON solver.

Effective quadratic cuts derived from structured LMIs.

Relaxations provide tight outer-approximations of the original problem.

Abstract

We consider the global optimization of nonconvex mixed-integer quadratic programs with linear equality constraints. In particular, we present a new class of convex quadratic relaxations which are derived via quadratic cuts. To construct these quadratic cuts, we solve a separation problem involving a linear matrix inequality with a special structure that allows the use of specialized solution algorithms. Our quadratic cuts are nonconvex, but define a convex feasible set when intersected with the equality constraints. We show that our relaxations are an outer-approximation of a semi-infinite convex program which under certain conditions is equivalent to a well-known semidefinite program relaxation. The new relaxations are implemented in the global optimization solver BARON, and tested by conducting numerical experiments on a large collection of problems. Results demonstrate that, for our test problems, these relaxations lead to a significant improvement in the performance of BARON.