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A remark on the lower semicontinuity assumption in the Ekeland variational principle

Xuan Duc Ha Truong (txdha***at***math.ac.vn)

Abstract: What happens to the conclusion of the Ekeland variational principle (briefly, EVP) if a considered function $f:X\to \R\cup\{+\infty\}$ is lower semicontinuous not on a whole metric space $X$ but only on its domain? We provide a straightforward proof showing that it still holds but only for $\epsilon $ varying in some interval $]0,\beta-\inf_Xf[$, where $\beta$ is a quantity expressing quantitatively the violation of the lower semicontinuity of $f$ outside its domain. This version of EVP collapses to the classical one when the function is lsc on the whole space.

Keywords: Ekeland variational principle, lower semicontinuity, G\^ateaux differentiability

Category 1: Nonlinear Optimization

Citation: OPTIMIZATION, 2016 VOL. 65, NO. 10, 17811789

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Entry Submitted: 07/09/2015
Entry Accepted: 07/09/2015
Entry Last Modified: 04/24/2017

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