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A version of the Ekeland variational principle for an extended real-valued function which is lower semicontinuous on its domain

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

Abstract: The well-known Ekeland variational principle (in brief, EVP) concerns a function $f:X\to\R\cup\{+\infty\}$, which is lower semicontinuous (in brief, lsc) and bounded from below on a complete metric space $X$. In this note, we answer the question: What happens when the assumption ``$f$ is lsc on $X$" is relaxed to ``$f$ is lsc on its domain"? We show that, in such a situation, the conclusion of EVP holds for $\epsilon$ varying in some interval $]0,\gamma -\inf_Xf[$. Here, $\gamma$ is the supremum of scalars $t$ with $t>\inf_Xf$ such that lower level sets $[f\leq t]$ are closed and it also is calculated based on information about the behavior of $f$ at the boundary of its domain. When $f$ is lsc on the whole space $X$, our version of EVP reduces to the classical one with $\gamma =+\infty$.

Keywords: Ekeland variational principle, lower semicontinuity, Gateaux differentiability

Category 1: Nonlinear Optimization (Other )


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Entry Submitted: 05/31/2015
Entry Accepted: 05/31/2015
Entry Last Modified: 05/31/2015

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