Compact mixed-integer programming relaxations in quadratic optimization

TL;DR

This paper introduces a novel mixed-integer programming relaxation technique for nonconvex quadratic optimization that uses a piecewise linear approximation, resulting in stronger bounds and improved computational performance.

Contribution

The paper develops a new MIP relaxation method leveraging a piecewise linear approximation with logarithmic complexity, enhancing bounds for nonconvex quadratic problems.

Findings

Outperforms existing MIP relaxations in literature

Produces tighter bounds than exact solvers on hard instances

Efficiently approximates nonconvex quadratic functions

Abstract

We present a technique for producing valid dual bounds for nonconvex quadratic optimization problems. The approach leverages an elegant piecewise linear approximation for univariate quadratic functions due to Yarotsky, formulating this (simple) approximation using mixed-integer programming (MIP). Notably, the number of constraints, binary variables, and auxiliary continuous variables used in this formulation grows logarithmically in the approximation error. Combining this with a diagonal perturbation technique to convert a nonseparable quadratic function into a separable one, we present a mixed-integer convex quadratic relaxation for nonconvex quadratic optimization problems. We study the strength (or sharpness) of our formulation and the tightness of its approximation. Further, we show that our formulation represents feasible points via a Gray code. We close with computational results on problems with quadratic objectives and/or constraints, showing that our proposed method i) across the board outperforms existing MIP relaxations from the literature, and ii) on hard instances produces better bounds than exact solvers within a fixed time budget.