Analysis of a Class of Minimization Problems Lacking Lower Semicontinuity

The minimization of non-lower semicontinuous functions is a difficult topic that has been minimally studied. Among such functions is a Heaviside composite function that is the composition of a Heaviside function with a possibly nonsmooth multivariate function. Unifying a statistical estimation problem with hierarchical selection of variables and a sample average approximation of composite chance … Read more

Continuous Selections of Solutions to Parametric Variational Inequalities

This paper studies the existence of a (Lipschitz) continuous (single-valued) solution function of parametric variational inequalities under functional and constraint perturbations. At the most elementary level, this issue can be explained from classical parametric linear programming and its resolution by the parametric simplex method, which computes a solution trajectory of the problem when the objective … Read more

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 $\beta$ … Read more

Analysis of direct searches for non-Lipschitzian functions

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