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

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.


OPTIMIZATION, 2016 VOL. 65, NO. 10, 1781–1789