Solving DC programs with a polyhedral component utilizing a multiple objective linear programming solver

TL;DR

This paper introduces a method for solving a class of non-convex DC optimization problems with polyhedral components by leveraging multiple objective linear programming solvers, demonstrated through numerical examples including locational analysis.

Contribution

It shows how to solve DC programs with polyhedral parts using MOLP solvers, connecting non-convex optimization to polyhedral projection and multi-objective linear programming.

Findings

Solution derived from polyhedral projection when g is polyhedral.

Solution obtained via polyhedral projection and convex programs when h is polyhedral.

Numerical examples demonstrate practical application, including locational analysis.

Abstract

A class of non-convex optimization problems with DC objective function is studied, where DC stands for being representable as the difference $f=g-h$ of two convex functions $g$ and $h$. In particular, we deal with the special case where one of the two convex functions $g$ or $h$ is polyhedral. In case $g$ is polyhedral, we show that a solution of the DC program can be obtained from a solution of an associated polyhedral projection problem. In case $h$ is polyhedral, we prove that a solution of the DC program can be obtained by solving a polyhedral projection problem and finitely many convex programs. Since polyhedral projection is equivalent to multiple objective linear programming (MOLP), a MOLP solver (in the second case together with a convex programming solver) can be used to solve instances of DC programs with polyhedral component. Numerical examples are provided, among them an application to locational analysis.