Algorithms to solve unbounded convex vector optimization problems

TL;DR

This paper introduces a new solution concept and algorithms for unbounded convex vector optimization problems, extending existing methods that only work for bounded problems by approximating recession cones.

Contribution

It proposes a generalized $( ext{ε,δ})$-solution concept and algorithms that handle unbounded CVOPs by bounding recession cones, filling a gap in current solution methods.

Findings

The algorithms successfully compute $( ext{ε,δ})$-solutions for unbounded CVOPs.

Numerical examples demonstrate the effectiveness of the proposed methods.

The approach extends polyhedral approximation techniques to unbounded problems.

Abstract

This paper is concerned with solution algorithms for general convex vector optimization problems (CVOPs). So far, solution concepts and approximation algorithms for solving CVOPs exist only for bounded problems [Ararat et al. 2022, Doerfler et al. 2021, Loehne et al. 2014]. They provide a polyhedral inner and outer approximation of the upper image that have a Hausdorff distance of at most $\varepsilon$. However, it is well known (see [Ulus, 2018]), that for some unbounded problems such polyhedral approximations do not exist. In this paper, we will propose a generalized solution concept, called an $(\varepsilon,\delta)$--solution, that allows also to consider unbounded CVOPs. It is based on additionally bounding the recession cones of the inner and outer polyhedral approximations of the upper image in a meaningful way. An algorithm is proposed that computes such $\delta$--outer and $\delta$--inner approximations of the recession cone of the upper image. In combination with the results of [Loehne et al. 2014] this provides a primal and a dual algorithm that allow to compute $(\varepsilon,\delta)$--solutions of (potentially unbounded) CVOPs. Numerical examples are provided.