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.
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Jahn, J., Ha, T.X.D. New Order Relations in Set Optimization. J Optim Theory Appl 148, 209–236 (2011). https://doi.org/10.1007/s10957-010-9752-8
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DOI: https://doi.org/10.1007/s10957-010-9752-8