A Hausdorff-type distance, a directional derivative of a set-valued map and applications in set optimization. (3rd July 2018)

• Record Type: Journal Article
• Authors: Ha, Truong Xuan Duc
• Is Part Of: Optimization. Volume 67:Number 7(2018)
• Page Start: 1031
• Page End: 1050
• Publication Date: 2018-07-03
• Subjects: Set-valued map -- directional derivative -- coderivative -- set optimization -- optimality condition; Mathematical optimization -- Periodicals; 519.7
• DOI: 10.1080/02331934.2017.1420186
• Languages: English
• ISSNs: 0233-1934

Abstract

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 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 .