Fix and Bound: An efficient approach for solving large-scale quadratic programming problems with box constraints

TL;DR

This paper introduces a branch-and-bound algorithm that combines SDP bounds and linear cuts to efficiently solve large-scale nonconvex quadratic programming problems with box constraints, significantly improving computational performance.

Contribution

The paper presents a novel branch-and-bound method integrating SDP bounds with optimality-based linear cuts for large-scale BoxQP problems, enabling variable fixing and size reduction.

Findings

State-of-the-art performance on large-scale BoxQPs up to 200 variables.

Competitive results on binary quadratic programming problems.

Effective reduction of problem size through variable fixing.

Abstract

In this paper, we propose a branch-and-bound algorithm for solving nonconvex quadratic programming problems with box constraints (BoxQP). Our approach combines existing tools, such as semidefinite programming (SDP) bounds strengthened through valid inequalities, with a new class of optimality-based linear cuts which leads to variable fixing. The most important effect of fixing the value of some variables is the size reduction along the branch-and-bound tree, allowing to compute bounds by solving SDPs of smaller dimension. Extensive computational experiments over large dimensional (up to $n=200$) test instances show that our method is the state-of-the-art solver on large-scale BoxQPs. Furthermore, we test the proposed approach on the class of binary QP problems, where it exhibits competitive performance with state-of-the-art solvers.