Covering properties at positive-order rates of multifunctions and some related topics

Abstract

We obtain equivalences among the covering property at a positive-order rate of a multifunction, the metric regularity property of a positive order, and the Hölder-like continuity property of the inverse mapping. Our results develop some aspects of the preceding results of J.-P. Penot, J.M. Borwein and D.M. Zhuang, H. Frankowska, B.S. Mordukhovich, and L.I. Minchenko. Necessary conditions for having these properties are given in terms of positive-order variational coderivative, a concept used here for the first time. We also discuss some known sufficient conditions for the validity of the three properties in terms of the positive-order variation in the sense of H. Frankowska. Illustrative examples are considered.

About[q]-regularity properties of collections of sets

“…In this section, we present relationships between [q]-regularity properties of collections of sets and the corresponding properties of set-valued mappings. Nonlinear regularity properties of set-valued mappings have been investigated, cf., e.g., [2,11,19,20,25,40,44,55].…”

Hölder Metric Subregularity with Applications to Proximal Point Method

“…Comprehensive characterizations of metric regularity and related well-posedness properties of set-valued mappings are available in the literature via both primal and dual constructions of generalized differentiation. Higher-order (Hölder) metric regularity and related properties of multifunctions have also been studied, e.g., in [9,13,35]. Note however that, while metric regularity is known to hold for a variety of constraint systems, it fails in typical variational frameworks; see [1,4,15,29] for more details and discussions.…”

Inclusions in general spaces: Hoelder stability, solution schemes and Ekeland's principle

“…Recently, conditions for the Aubin property [q > 0] -similar as in [44] -were given by means of modified openness in [52]. There, also generalized derivatives as in [4,17,18] have been considered, and it has been shown that a modified coderivative is necessarily injective under that property.…”