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A generalized distance and enhanced Ekeland’s variational principle for vector functions. (English) Zbl 1194.58014
Summary: We propose a definition of lower closed transitive relations and prove the existence of minimal elements for such a relation. This result is shown to contain probably a large part of existing versions of Ekeland’s variational principle (EVP). We introduce the notion of a weak $\tau$-function $p$ as a generalized distance and use it together with the above result on minimal elements to establish enhanced EVP for various settings, under relaxed lower semicontinuity assumptions. These principles conclude the existence not only of $p$-strict minimizers of $p$-perturbations of the considered vector function, but also $p$-sharp and $p$-strong minimizers. Our results are proved to be stronger than the classical EVP and many generalizations in the literature, even in the usual one-dimensional case, by numerous corollaries and examples. We include equivalent formulations of our enhanced EVP as well.

##### MSC:
 58E30 Variational principles on infinite-dimensional spaces 65K10 Optimization techniques (numerical methods) 49J53 Set-valued and variational analysis 90C48 Programming in abstract spaces