NDLI: New Order Relations in Set Optimization

Content Provider	Springer Nature Link
Author	Jahn, Johannes Ha, Truong Xuan Duc
Copyright Year	2010
Abstract	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 $\mathcal{P}(Y)$ 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 $(\mathcal{Q},\preccurlyeq)$ without any topological or linear structure. Namely, we define the following concepts w.r.t. the pre-order $\preccurlyeq$ : minimal elements, semicompactness, completeness, domination property of a subset of $\mathcal{Q}$ , and semicontinuity of a set-valued map with values in $\mathcal{Q}$ in a topological setting. Secondly, we establish existence results for optimal solutions of (SOP), when F takes values on $(\mathcal{Q},\preccurlyeq)$ from which one can easily derive similar results for the case, when F takes values on $\mathcal{P}(Y)$ equipped with various order relations.
Starting Page	209
Ending Page	236
Page Count	28
File Format	PDF
ISSN	00223239
Journal	Journal of Optimization Theory and Applications
Volume Number	148
Issue Number	2
e-ISSN	15732878
Language	English
Publisher	Springer US
Publisher Date	2010-10-16
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Set optimization Order relations Existence results Engineering Applications of Mathematics Optimization Calculus of Variations and Optimal Control; Optimization Theory of Computation Operations Research/Decision Theory
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Control and Optimization Management Science and Operations Research

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