Algorithms to solve unbounded convex vector optimization 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 [\c C. Ararat, F. Ulus, and M. Umer, J. Optim. Theory Appl., 194 (2022), pp. 681-712], [D. Dörfler, A. Löhne, C. Schneider, and B. Weißing, Optim. Methods Softw., 37 (2022), pp. 1006-1026], [A. Löhne, B. Rudloff, and F. Ulus, J. Global Optim., 60 (2014), pp. 713-736]. They provide a polyhedral inner and outer approximation of the upper image that have a Hausdorff distance of at most ε. However, it is well known (see [F. Ulus, J. Global Optim., 72 (2018), pp. 731-742]), that for some unbounded problems such polyhedral approximations do not exist. In this paper, we will propose a generalized solution concept, called an (ε,δ)-solution, that allows one to also 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 δ-outer and δ-inner approximations of the recession cone of the upper image. In combination with the results of [A. Löhne, B. Rudloff, and F. Ulus, J. Global Optim., 60 (2014), pp. 713-736] this provides a primal and a dual algorithm that allow one to compute (ε,δ)-solutions of (potentially unbounded) CVOPs. Numerical examples are provided.