Proper minimal points of nonconvex sets in Banach spaces in terms of the limiting normal cone. (3rd April 2017)

• Record Type: Journal Article
• Authors: Shahbeyk, Shokouh; Soleimani-damaneh, Majid
• Is Part Of: Optimization. Volume 66:Number 4(2017)
• Page Start: 473
• Page End: 489
• Publication Date: 2017-04-03
• Subjects: Proper minimal point -- limiting (Mordukhovich) normal cone -- nonsmooth analysis -- ordered Banach space -- Asplund space; Mathematical optimization -- Periodicals; 519.7
• DOI: 10.1080/02331934.2017.1289528
• Languages: English
• ISSNs: 0233-1934

Abstract

Abstract: In this paper, proper minimal elements of a given nonconvex set in a real ordered Banach space are defined utilizing the limiting (Mordukhovich) normal cone. The newly defined points are called limiting proper minimal (LPM) points. It is proved that each LPM is a proper minimal in the sense of Borwein under some assumptions. The converse holds in Asplund spaces. The relation of LPM points with Benson, Henig, super and proximal proper minimal points are established. Under appropriate assumptions, it is proved that the set of robust elements is a subset of the set of LPM points, and the set of LPM points is dense in that of minimal points. Another part of the paper is devoted to scalarization-based and distance function-based characterizations of the LPM points. The paper is closed by some results about LPM solutions of a set-valued optimization problem via variational analysis tools. Clarifying examples are given in addition to the theoretical results.