A Hausdorff-type distance,a directional derivative of a set-valued map and applications in set optimization |
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Authors: | Truong Xuan Duc Ha |
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Institution: | 1. Institute of Mathematics, Vietnam Academy of Sciences and Technology , Hanoi, Viet Nam.txdha@math.ac.vn |
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Abstract: | AbstractIn this paper, we follow Kuroiwa’s set approach in set optimization, which proposes to compare values of a set-valued objective map F with respect to various set order relations. We introduce a Hausdorff-type distance relative to an ordering cone between two sets in a Banach space and use it to define a directional derivative for F. We show that the distance has nice properties regarding set order relations and the directional derivative enjoys most properties of the one of a scalar single-valued function. These properties allow us to derive necessary and/or sufficient conditions for various types of maximizers and minimizers of F. |
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Keywords: | Set-valued map directional derivative coderivative set optimization optimality condition |
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