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.