Nonsmooth sequential analysis in Asplund spaces

Abstract

Abstract. We develop a generalized differentiation theory for nonsmooth functions and sets with nonsmooth boundaries defined in Asplund spaces. This broad subclass of Banach spaces provides a convenient framework for many important applications to optimization, sensitivity, variational inequalities, etc. Our basic normal and subdifferential constructions are related to sequential weak-star limits of Fréchet normals and subdifferentials. Using a variational approach, we establish a rich calculus for these nonconvex limiting objects which turn out to be minimal among other set-valued differential constructions with natural properties. The results obtained provide new developments in infinite dimensional nonsmooth analysis and have useful applications to optimization and the geometry of Banach spaces.

Necessary Optimality Conditions for Control of Strongly Monotone Variational Inequalities

“…As proved in Mordukhovich and Shao (1996a), in Asplund spaces X the normal cone (1.5) admits the simplified representation…”

“…is Lipschitz at (y, u, 0). Hence it is directionally Lipschitz by Theorem 2.9.4 of Clarke (1983) and 8 00 g(Y, u, 0) = {0} by Proposition 2.5 of Mordukhovich and Shao (1996a).…”

Are Generalized Derivatives Sseful for Generalized Convex Functions?

“…This is the case for the approximate subdifferential ( [82]- [86]), the limiting subdifferential ( [136]- [138]), the proximal subdifferential, the viscosity subdifferential ( [5]) and the firm (or Frechet) subdilferential. The last one is given by Among the additional properties which may be satisfied is the following one.…”

“…Still, since coderivatives are the adapted tool for studying multimappings (correspondences or relations) they certainly have a role to play in a field in which the sublevel set multimapping r t-t F(r) := [/ ~ r} := /-1 ()-00, r]) associated with a function / plays a role which is more important than the role played by the epigraph of /. An explanation also lies in the freshness of the subject (see [86], [87], [138], [154) and their references for instance).…”

On nonconvex subdifferential calculus in binormed spaces

“…In the beginning Ioffe, Jourani, kruger, Loewen, Mordukhovich, Shao and Thibault used the limiting subdifferential to establish full principal calculus rules and applications in Banach spaces which possess some geometric assumptions (we can see [2,3,4,9,12,13]). Later, Mordukhovich and Shao in [11] extend some of those results to arbitrary Banach spaces including sum rules, chain rules, product and quotient rules, however most of those results must involve the strict differentiability notion.…”