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A Hausdorff-type distance, a directional derivative of a set-valued map and applications in set optimization

Xuan Duc Ha Truong (txdha***at***math.ac.vn)

Abstract: In this paper, we follow Kuroiwa's set approach in set optimization, which proposes to compare values of a set-valued objective map $F$ 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 $F$.

Keywords: Set-valued map, directional derivative, coderivative, set optimization, optimality condition

Category 1: Other Topics (Multi-Criteria Optimization )

Category 2: Convex and Nonsmooth Optimization (Generalized Convexity/Monoticity )

Citation: Optimization(2018) Vol. 67, No.7, 1031-1050


Entry Submitted: 04/23/2017
Entry Accepted: 04/23/2017
Entry Last Modified: 07/19/2018

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