Minimal element theorems and Ekeland's principle with set relations
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SubjectSet relations; Set-valued variational principle; Minimal point theorem; Set-valued optimization
We present two existence principles for minimal points of subsets of the product space X × 2Y, where X stands for a separated uniform space and Y a topological vector space. The two principles are distinct with respect to the involved ordering structure in 2Y. We derive from them new variants of Ekeland's principle for set-valued maps as well as a minimal point theorem in X × Y and Ekeland's principle for vector-valued functions.
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Weak minimizers, minimizers and variational inequalities for set-valued functions: A blooming wreath? Crespi GP; Schrage C (2016)Recently, necessary and sufficient conditions in terms of variational inequalities have been introduced to characterize minimizers of convex setvalued functions. Similar results have been proved for a weaker concept of ...
Hamel, AH (TAYLOR & FRANCIS LTD, 2011)A duality theorem of the Fenchel-Rockafellar type for set-valued optimization problems is presented along with a result for the conjugate of the sum of two set-valued functions and a chain rule. The underlying solution ...
Hamel, AH (2003)A generalization of Phelps' lemma to locally convex spaces is proven, applying its well-known Banach space version. We show the equivalence of this theorem, Ekeland's principle and Danes' drop theorem in locally convex ...