
Minimax Theory and its Applications 09 (2024), No. 2, 271286 Copyright Heldermann Verlag 2024 On Unbounded Polyhedral Convex Set Optimization Problems Niklas Hey Institute for Statistics and Mathematics, University of Economics and Business, Vienna, Austria niklas.hey@wu.ac.at Andreas Löhne Faculty of Mathematics and Computer Science, Friedrich Schiller University, Jena, Germany andreas.loehne@unijena.de A polyhedral convex set optimization problem is given by a setvalued objective mapping from the ndimensional to the qdimensional 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 scalarvalued objective function. Polyhedral convex set optimization generalizes both scalar and multiobjective (or vector) linear programming. In contrast to scalar linear programming but likewise to multiobjective 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 multiobjective 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 by A. Löhne and C. Schrage [An algorithm to solve polyhedral convex set optimization problems, Optimization 62/1 (2013) 131141] to the case of not necessarily bounded problems uncovers some problems, which will be discussed. Keywords: Set optimization, solution methods, vector linear programming, multiple objective linear programming. MSC: 90C99, 90C29, 90C05. [ Fulltextpdf (128 KB)] for subscribers only. 