New Order Relations in Set Optimization

• Published in: Journal of optimization theory and applications Vol. 148; no. 2; pp. 209 - 236
• Main Authors: Jahn, Johannes, Ha, Truong Xuan Duc
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.02.2011; Springer; Springer Nature B.V
• Subjects: Applications of Mathematics; Applied sciences; Calculus of Variations and Optimal Control; Optimization; Completeness; Engineering; Exact sciences and technology; Mathematical analysis; Mathematical programming; Mathematics; Mathematics and Statistics; Operational research and scientific management; Operational research. Management science; Operations Research/Decision Theory; Optimization; Studies; Theory of Computation; Topology; Vectors (mathematics); Existence results; Order relations; Set optimization; Cone; Multivalued function; Existence theorem; Ordered set; Linear structure; Dominance; Topology; Topological structure; Graph decomposition; Completeness; Optimal solution; Convexity; Semicontinuous map; Convex analysis; Vector optimization
• ISSN: 0022-3239; 1573-2878
• DOI: 10.1007/s10957-010-9752-8

In this paper we study a set optimization problem ( SOP ), i.e. we minimize a set-valued objective map F , which takes values on a real linear space Y equipped with a pre-order induced by a convex cone  K . We introduce new order relations on the power set of Y (or on a subset of it), which are more suitable from a practical point of view than the often used minimizers in set optimization. Next, we propose a simple two-steps unifying approach to studying ( SOP ) w.r.t. various order relations. Firstly, we extend in a unified scheme some basic concepts of vector optimization, which are defined on the space Y up to an arbitrary nonempty pre-ordered set without any topological or linear structure. Namely, we define the following concepts w.r.t. the pre-order : minimal elements, semicompactness, completeness, domination property of a subset of  , and semicontinuity of a set-valued map with values in in a topological setting. Secondly, we establish existence results for optimal solutions of ( SOP ), when F takes values on from which one can easily derive similar results for the case, when F takes values on equipped with various order relations.