Cutting Plane Algorithms are Exact for Euclidean Max-Sum Problems

TL;DR

This paper introduces two exact cutting plane algorithms for Euclidean max-sum problems, efficiently solving large-scale binary quadratic programs with polyhedral constraints without relying on slow concave reformulations.

Contribution

The paper presents novel cutting plane algorithms that directly address Euclidean max-sum problems, improving convergence speed and scalability over existing methods.

Findings

Algorithms outperform existing methods on benchmark problems

Can solve problems with up to 3000 variables

Remove the need for slow concave reformulations

Abstract

This paper studies binary quadratic programs in which the objective is defined by a Euclidean distance matrix, subject to a general polyhedral constraint set. This class of nonconcave maximisation problems includes the capacitated, generalised and bi-level diversity problems as special cases. We introduce two exact cutting plane algorithms to solve this class of optimisation problems. The new algorithms remove the need for a concave reformulation, which is known to significantly slow down convergence. We establish exactness of the new algorithms by examining the concavity of the quadratic objective in a given direction, a concept we refer to as directional concavity. Numerical results show that the algorithms outperform other exact methods for benchmark diversity problems (capacitated, generalised and bi-level), and can easily solve problems of up to three thousand variables.