An exact cutting plane method for solving p-dispersion-sum problems

TL;DR

This paper introduces an exact cutting plane method for the p-dispersion-sum problem, leveraging the conditionally negative definite Euclidean distance matrix to efficiently find optimal solutions, outperforming existing methods especially on large instances.

Contribution

It demonstrates that the cutting plane method is exact for PDSP due to the matrix's properties, solving large instances efficiently and filling a gap in the literature.

Findings

Method outperforms existing exact algorithms.

Can solve large instances with up to 2000 variables.

Converges to the optimal solution despite non-concavity.

Abstract

This paper aims to answer an open question recently posed in the literature, that is to find a fast exact method for solving the p-dispersion-sum problem (PDSP), a nonconcave quadratic binary maximization problem. We show that, since the Euclidean distance matrix defining the quadratic term in (PDSP) is always conditionally negative definite, the cutting plane method is exact for (PDSP) even in the absence of concavity. As such, the cutting plane method, which is primarily designed for concave maximisation problems, converges to the optimal solution of the (PDSP). The numerical results show that the method outperforms other exact methods for solving (PDSP), and can solve to optimality large instances of up to two thousand variables.