Jahn J, Ha T (2011)
Publication Type: Journal article, Original article
Publication year: 2011
Publisher: Springer Verlag (Germany)
Book Volume: 148
Pages Range: 209-236
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 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(Q) 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 Q, and semicontinuity of a set-valued map with values in Q in a topological setting. Secondly, we establish existence results for optimal solutions of (SOP), when F takes values on(Q) from which one can easily derive similar results for the case, when F takes values on P(Y) equipped with various order relations. © 2010 Springer Science+Business Media, LLC.
APA:
Jahn, J., & Ha, T. (2011). New Order Relations in Set Optimization. Journal of Optimization Theory and Applications, 148, 209-236. https://dx.doi.org/10.1007/s10957-010-9752-8
MLA:
Jahn, Johannes, and T.X.D. Ha. "New Order Relations in Set Optimization." Journal of Optimization Theory and Applications 148 (2011): 209-236.
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