Ekeland variational principles for set-valued functions with set perturbations

Abstract

By using generalized nonconvex separation functionals and a pre-order principle in Qiu [J Math Anal Appl. 2014;419:904–937], we establish a general set-valued Ekeland variational principle (briefly, denoted by EVP), where the objective function is a set-valued map taking values in a real vector space quasi-ordered by a convex cone K and the perturbation consists of a cone-convex subset H of K multiplied by the distance function. Here, the assumption on lower semi-continuity of the objective function is replaced by a weaker one: sequentially lower monotony. And the assumption on lower boundedness of the objective function is taken to be the weakest of several different kinds. From the general set-valued EVP, we deduce a number of particular versions of set-valued EVP, which extend and improve the related results in the literature. In particular, we give several EVPs for approximately efficient solutions in set-valued optimization, which not only extend the related results from vector-valued objective functions into set-valued objective functions, but also improve the related results by removing a usual assumption for K-boundedness (by scalarization) of the objective function's range.

Keywords:

• Ekeland variational principle
• set-valued objective function
• set perturbation
• nonconvex separation functional
• vector closure
• ($C,ϵ$)-efficient solution

AMS subject classifications:

• 58E30
• 49J53
• 65K10
• 90C26
• 90C48

Acknowledgments

The authors are grateful to the reviewers for the favourable comments and helpful suggestions, which have greatly improved our paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11471236, 11561049).