Extended Triangle Inequalities for Nonconvex Box-Constrained Quadratic Programming

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.