# Solving polyhedral d.c. optimization problems via concave minimization

## TL;DR

This paper characterizes conditions for global optimal solutions in polyhedral difference-of-convex (d.c.) optimization problems and demonstrates how these problems can be solved using concave minimization algorithms, with practical tests and procedures.

## Contribution

It provides a characterization of solution existence and introduces solution methods for polyhedral d.c. problems using concave minimization, including primal and dual tests.

## Key findings

- Existence of solutions can be certified under certain conditions.
- Polyhedral d.c. problems can be solved with concave minimization algorithms.
- Primal and dual solution procedures are developed for cases with both functions polyhedral.

## Abstract

The problem of minimizing the difference of two convex functions is called polyhedral d.c. optimization problem if at least one of the two component functions is polyhedral. We characterize the existence of global optimal solutions of polyhedral d.c. optimization problems. This result is used to show that, whenever the existence of an optimal solution can be certified, polyhedral d.c. optimization problems can be solved by certain concave minimization algorithms. No further assumptions are necessary in case of the first component being polyhedral and just some mild assumptions to the first component are required for the case where the second component is polyhedral. In case of both component functions being polyhedral, we obtain a primal and dual existence test and a primal and dual solution procedure. Numerical examples are discussed.