Extended Triangle Inequalities for Nonconvex Box-Constrained Quadratic Programming

TL;DR

This paper introduces new linear inequalities and conic strengthenings to better characterize the convex hull of nonconvex quadratic problems over box constraints, improving solution accuracy for certain instances.

Contribution

The paper develops systematic valid inequalities and conic relaxations that enhance the approximation of the convex hull of nonconvex quadratic programs over box constraints.

Findings

New inequalities improve solution accuracy for some nonconvex instances.

Conic relaxations incorporate second-order cone constraints for better bounds.

Exact solutions achieved for instances unsolved by previous constraints.

Abstract

Let $\rm{Box}_n = \{x \in \mathbb{R}^n : 0 \leq x \leq e \}$, and let $\rm{QPB}_n$ denote the convex hull of $\{(1, x')'(1, x') : x \in \rm{Box}_n\}$. The quadratic programming problem $\min\{x'Q x + q'x : x \in \rm{Box}_n\}$ where $Q$ is not positive semidefinite (PSD), is equivalent to a linear optimization problem over $\rm{QPB}_n$ and could be efficiently solved if a tractable characterization of $\rm{QPB}_n$ was available. It is known that $\rm{QPB}_2$ can be represented using a PSD constraint combined with constraints generated using the reformulation-linearization technique (RLT). The triangle (TRI) inequalities are also valid for $\rm{QPB}_3$, but the PSD, RLT and TRI constraints together do not fully characterize $\rm{QPB}_3$. In this paper we describe new valid linear inequalities for $\rm{QPB}_n$, $n \geq 3$ based on strengthening the approximation of $\rm{QPB}_3$ given by the PSD, RLT and TRI constraints. These new inequalities are generated in a systematic way using a known disjunctive characterization for $\rm{QPB}_3$. We also describe a conic strengthening of the linear inequalities that incorporates second-order cone constraints. We show computationally that the new inequalities and their conic strengthenings obtain exact solutions for some nonconvex box-constrained instances that are not solved exactly using the PSD, RLT and TRI constraints.