Algorithms and dimensionality reductions for continuous multifacility ordered median location problems

TL;DR

This paper introduces new optimization methodologies for continuous multifacility ordered median location problems using second order cone programming and semidefinite programming, incorporating dimensionality reduction techniques for larger problem instances.

Contribution

It presents a novel general approach combining second order cone mixed integer programming and semidefinite programming for these location problems, with techniques for dimensionality reduction.

Findings

New second order cone mixed integer programming formulation

Sequence of semidefinite relaxations converging to the solution

Dimensionality reduction methods enabling larger problem solving

Abstract

In this paper we propose a general methodology for solving a broad class of continuous, multifacility location problems, in any dimension and with $\ell_\tau$-norms proposing two different methodologies: 1) by a new second order cone mixed integer programming formulation and 2) by formulating a sequence of semidefinite programs that converges to the solution of the problem; each of these relaxed problems solvable with SDP solvers in polynomial time. We apply dimensionality reductions of the problems by sparsity and symmetry in order to be able to solve larger problems.