A Set-Valued Ekeland's Variational Principle in Vector Optimization

Ekeland's Variational Principle for Set-Valued Functions

“…Thus, instead of approaches of earlier authors asserting relations between elements of the values of the set-valued function F , our extension (reported in Theorem 3.4 and Theorem 3.5) of the Ekeland principle are directly expressed in terms of values (as entities) of F . The idea of replacing a singleton by a set or a suitable set-valued function has very recently been used by Bednarczuk-Zagrodny [5] and Gutiérrez-Jiménez-Novo [15] in their extended Ekeland's variational principles for vector-valued (single-valued) functions. Our discussion on the issue of error bounds in the next section will further shed light on using a set H instead of a singleton in our extensions of the Ekeland's variational principle.…”

“…The authors would like to thank Prof. C. C. Chou for his stimulating discussion, and thank referees for their constructive comments leading to many improvements and for pointing out the references [3,5,15,16,21,22,38].…”

“…There are many papers (see [6,8,11,35] for example) reporting different formulations and some have put forward extended versions applicable to vector-valued/set-valued functions (see [3,4,5,9,15,16,18,38]). This paper is devoted to further extensions which are especially relevant for the study of error bound issue for the following inequality system:…”

The Ekeland variational principle for Henig proper minimizers and super minimizers

“…During the last three decades many authors have been interested in extending EVP and related results to a single-or setvalued map with values in a vector space partially ordered by a convex cone, see [4,5,[7][8][9]11,13,14,[16][17][18][19][20][21][23][24][25]28,29,32,33,35,38,46,47,49,53] and the references therein. A brief enquiry into the matter reveals that the authors concentrated mainly on exact/approximate Pareto minimizers or exact/approximate weak minimizers, and the ordering cone was often assumed to have a nonempty interior.…”

Optimality Conditions for Approximate Solutions in Multiobjective Optimization Problems

“…In vector optimization, the notion of approximate solution has been defined in several ways. The first concept was introduced by Kutateladze 10 and has been used to establish vector variational principle, approximate Kuhn-Tucker-type conditions and approximate duality theorems, and so forth, see [11][12][13][14][15][16][17][18][19][20] . Later, several authors have proposed other -efficiency concepts see, e.g., White 21 ; Helbig 22 and Tanaka 23 .…”