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Extended Triangle Inequalities for Nonconvex Box-Constrained Quadratic Programming

Kurt M. Anstreicher, Diane Puges

TL;DR

This work tackles nonconvex box-constrained quadratic programming by studying the convex hull $QPB_n$ and tightening its standard relaxations. It introduces extended triangle inequalities ($ETRI$) for ${QPB}_3$ via a disjunctive triangulation, then strengthens them with $ETRI2$ and $ETRI3$ using eigenvector-based cuts, and further provides a conic strengthening based on second-order-cone constraints. The combined approach yields significantly tighter bounds, achieving exact solutions on several challenging ${QPB}_3$ instances (notably the Burer–Letchford example) and improving gaps on larger instances with $n$ up to 10, while remaining tractable in practice due to a small number of active inequalities. The results demonstrate that the $ETRI$ framework, especially with the SOC enhancements, meaningfully extends the reach of convex relaxations for nonconvex box-constrained quadratic problems, reducing reliance on branching or more complex global formulations.

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.

Extended Triangle Inequalities for Nonconvex Box-Constrained Quadratic Programming

TL;DR

This work tackles nonconvex box-constrained quadratic programming by studying the convex hull and tightening its standard relaxations. It introduces extended triangle inequalities () for via a disjunctive triangulation, then strengthens them with and using eigenvector-based cuts, and further provides a conic strengthening based on second-order-cone constraints. The combined approach yields significantly tighter bounds, achieving exact solutions on several challenging instances (notably the Burer–Letchford example) and improving gaps on larger instances with up to 10, while remaining tractable in practice due to a small number of active inequalities. The results demonstrate that the framework, especially with the SOC enhancements, meaningfully extends the reach of convex relaxations for nonconvex box-constrained quadratic problems, reducing reliance on branching or more complex global formulations.

Abstract

Let , and let denote the convex hull of . The quadratic programming problem where is not positive semidefinite (PSD), is equivalent to a linear optimization problem over and could be efficiently solved if a tractable characterization of was available. It is known that 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 , but the PSD, RLT and TRI constraints together do not fully characterize . In this paper we describe new valid linear inequalities for , based on strengthening the approximation of given by the PSD, RLT and TRI constraints. These new inequalities are generated in a systematic way using a known disjunctive characterization for . 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.
Paper Structure (5 sections, 5 theorems, 33 equations, 9 tables)

This paper contains 5 sections, 5 theorems, 33 equations, 9 tables.

Table of Contents

  1. Introduction
  2. Extended triangle inequalities
  3. More extended triangle inequalities
  4. Conic strengthening
  5. Computational results

Key Result

Lemma 1

The set of $\{x_1, x_2, x_3,X_{11}, X_{22}, X_{33},X_{12}, X_{13}, X_{23}\}\subset\mathbb{R}^9$ with $(x,X)\in {\rm QPB}_3$ that also satisfy $2x_1 + X_{11} - 2X_{12} - 2X_{13} + X_{23}=0$ has dimension 5.

Theorems & Definitions (10)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof