An exact copositive programming formulation for the Discrete Ordered Median Problem: Extended version

TL;DR

This paper introduces a novel continuous linear conic formulation for the Discrete Ordered Median Problem, enabling the application of continuous optimization techniques to a traditionally combinatorial problem.

Contribution

It provides the first exact continuous convex formulation for DOMP by transforming a quadratic binary model into a linear conic problem involving completely positive matrices.

Findings

Establishes equivalence between DOMP and a continuous convex problem.

Enables new optimization approaches for DOMP using continuous methods.

Provides theoretical insights into the structure of ordered median problems.

Abstract

This paper presents a first continuous, linear, conic formulation for the Discrete Ordered Median Problem (DOMP). Starting from a binary, quadratic formulation in the original space of location and allocation variables that are common in Location Analysis (L.A.), we prove that there exists a transformation of that formulation, using the same space of variables, that allows us to cast DOMP as a continuous linear problem in the space of completely positive matrices. This is the first positive result that states equivalence between the family of continuous convex problems and some hard problems in L.A. The result is of theoretical interest because it allows us to share the tools from continuous optimization to shed new light into the difficult combinatorial structure of the class of ordered median problems.