New Order Relations in Set Optimization

Johannes Jahn () and Truong Xuan Duc Ha ()
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Johannes Jahn: Universität Erlangen-Nürnberg
Truong Xuan Duc Ha: Institute of Mathematics

Journal of Optimization Theory and Applications, 2011, vol. 148, issue 2, No 1, 209-236

Abstract: 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.

Keywords: Set optimization; Order relations; Existence results (search for similar items in EconPapers)
Date: 2011
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Citations: View citations in EconPapers (9)

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DOI: 10.1007/s10957-010-9752-8

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