lower semicontinuity – Optimization Online

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

Published: 2015/07/09, Updated: 2017/04/24

Nonlinear Optimization ekeland variational principle, gateaux differentiability, lower semicontinuity

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 $\beta$ … Read more

Analysis of direct searches for non-Lipschitzian functions

Published: 2009/10/12

Ana Luisa Custódio

Luis Nunes Vicente

Constrained Nonlinear Optimization, Nonlinear Optimization direct search, directionally lipschitzian functions, discontinuities, lipschitz extensions, lower semicontinuity, nonsmooth calculus, rockafellar directional derivatives

It is known that the Clarke generalized directional derivative is nonnegative along the limit directions generated by directional direct-search methods at a limit point of certain subsequences of unsuccessful iterates, if the function being minimized is Lipschitz continuous near the limit point. In this paper we generalize this result for non-Lipschitzian functions using Rockafellar generalized … Read more