On unbounded polyhedral convex set optimization problems

TL;DR

This paper extends the concept of finite infimizers to unbounded polyhedral convex set optimization problems, linking solutions to vector linear reformulations and exploring new solution concepts within a lattice framework.

Contribution

It generalizes finite infimizers to unbounded problems and connects solutions to vector linear reformulations, advancing the theory of set optimization.

Findings

Finite infimizers can be derived from vector linear reformulations.

Unbounded problems may still have minimizers, unlike scalar linear programming.

Extensions of solution concepts reveal new challenges in unbounded set optimization.

Abstract

A polyhedral convex set optimization problem is given by a set-valued objective mapping from the $n$-dimensional to the $q$-dimensional Euclidean space whose graph is a convex polyhedron. This problem can be seen as the most elementary subclass of set optimization problems, comparable to linear programming in the framework of optimization with scalar-valued objective function. Polyhedral convex set optimization generalizes both scalar and multi-objective (or vector) linear programming. In contrast to scalar linear programming but likewise to multi-objective linear programming, unbounded problems can indeed have minimizers and provide a rich class of problem instances. In this paper we extend the concept of finite infimizers from multi-objective linear programming to not necessarily bounded polyhedral convex set optimization problems. We show that finite infimizers can be obtained from finite infimizers of a reformulation of the polyhedral convex set optimization problem into a vector linear program. We also discuss two natural extensions of solution concepts based on the complete lattice approach. Surprisingly, the attempt to generalize the solution procedure for bounded polyhedral convex set optimization problems introduced in [A. L\"ohne and C. Schrage. An algorithm to solve polyhedral convex set optimization problems. {\em Optimization} 62(1):131--141, 2013.] to the case of not necessarily bounded problems uncovers some problems, which will be discussed.