Ekeland variational principles with set-valued objective functions and set-valued perturbations

TL;DR

This paper develops a comprehensive set-valued Ekeland variational principle in real vector spaces with minimal boundedness assumptions, extending existing results and applying to approximate solutions in set-valued optimization.

Contribution

It introduces a general set-valued EVP with weak lower boundedness, extending prior results and covering new cases in set-valued optimization and quasi-ordered spaces.

Findings

Established a general set-valued EVP with minimal boundedness assumptions.

Derived multiple specific EVPs for approximate efficient solutions.

Extended the EVP framework to quasi-ordered topological vector spaces.

Abstract

In the setting of real vector spaces, 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 $K$-convex subset $H$ of the ordering cone $K$ multiplied by the distance function. Here, the assumption on lower boundedness of the objective function is taken to be the weakest kind. 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, where a usual assumption for $K$-boundedness (by scalarization) of the objective function's range is removed. Moreover, still under the weakest lower boundedness condition, we present a set-valued EVP, where the objective function is a set-valued map taking values in a quasi-ordered topological vector space and the perturbation consists of a $\sigma$-convex subset of the ordering cone multiplied by the distance function.