Calculus of directional subdifferentials and coderivatives in Banach spaces

In this work we study the directional versions of Mordukhovich normal cones to nonsmooth sets, coderivatives of set-valued mappings, and subdifferentials of extended-real-valued functions in the framework of general Banach spaces. We establish some characterizations and basic properties of these constructions, and then develop calculus including sum rules and chain rules involving smooth functions. As an application, we also explore the upper estimates of the directional Mordukhovich subdifferentials and singular subdifferentials of marginal functions.     

Generalized sequential normal compactness in Asplund spaces

Sequential normal compactness conditions are important properties in infinite-dimensional variational analysis and its applications. Following the recent study of the generalized sequential normal compactness (GSNC), this paper This paper reveals further applications of GSNC to the generalized differentiation theory in Asplund spaces, as well as the calculus of GSNC itself.     

Universality of Asplund spaces

Given any infinite cardinal , there exists no Banach space of density , which is Asplund or has the Point of Continuity Property and is universal for all reflexive spaces of density .

     

Boundaries of Asplund spaces

We study the relationship between the classical combinatorial inequalities of Simons and the more recent (I)-property of Fonf and Lindenstrauss. We obtain a characterization of strong boundaries for Asplund spaces using the new concept of finitely self-predictable set. Strong properties for w^∗-K-analytic boundaries are established as well as a sup-lim sup theorem for Baire maps.     

Subdifferential calculus in Asplund generated spaces

We extend the definition of the limiting Fréchet subdifferential and the limiting Fréchet normal cone from Asplund spaces to Asplund generated spaces. Then we prove a sum rule, a mean value theorem, and other statements for this concept.     

Nonsmooth sequential analysis in Asplund spaces

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.

     

On generalized convexity in Asplund spaces

In this paper some properties of nonsmooth quasiconvex and pseudoconvex functions in weakly compactly generated Asplund spaces are proved.     

On subclasses of weak Asplund spaces

Assuming the consistency of the existence of a measurable cardinal, it is consistent to have two Banach spaces, , where is a weak Asplund space such that (in the weak* topology) in not in Stegall's class, whereas is in Stegall's class but is not weak* fragmentable.

     

A mean value theorem in Asplund spaces

In this paper a nonsmooth mean value theorem in Asplund spaces, under invexity, is provided.     

Weak regularity of functions and sets in Asplund spaces

In this paper, we study a new concept of weak regularity of functions and sets in Asplund spaces. We show that this notion includes prox-regular functions, functions whose subdifferential is weakly submonotone and amenable functions in infinite dimension. We establish also that weak regularity is equivalent to Mordukhovich regularity in finite dimension. Finally, we give characterizations of the weak regularity of epi-Lipschitzian sets in terms of their local representations.     

Extensions of generalized differential calculus in Asplund spaces

We develop an extended generalized differential calculus for normal cones to nonconvex sets, coderivatives of set-valued mappings, and subdifferential of extended-real-valued functions in infinite dimensions. This is a major area of modern variational analysis important for many applications, particularly to optimization, sensitivity, and control. We develop a unified geometric approach to the generalized differential calculus and obtain new results in this direction in a broad setting of Asplund spaces.     

The duality between Asplund spaces and spaces with the Radon-Nikodym property

A Banach spaceX is an Asplund space (a strong differentiability space) if and only ifX ^* has the Radon-Nikodym property.     

Calmness and inverse image characterizations for Asplund spaces

We establish new characterizations of Asplund spaces in terms of conditions ensuring the calmness property for constraint set mappings and the validity of inverse image formula for a general constrained system.     

Chain rules for coderivatives of multivalued mappings in Banach spaces

Our basic object in this paper is to establish calculus rules for coderivatives of multivalued mappings between Banach spaces. We consider the coderivative which is associated to some geometrical approximate subdifferential for functions.

     

Vector variational-like inequalities and vector optimization problems in Asplund spaces

In this paper, the Minty vector variational-like inequality, the Stampacchia vector variational-like inequality, and the weak formulations of these two inequalities defined by means of Mordukhovich limiting subdifferentials are introduced and studied in Asplund spaces. Some relations between the vector variational-like inequalities and vector optimization problems are established by using the properties of Mordukhovich limiting subdifferentials. An existence theorem of solutions for the weak Minty vector variational-like inequality is also given.     

Characterizations and applications of generalized invexity and monotonicity in Asplund spaces

In this paper, the concepts of invexity, monotonicity, and their generalizations in Asplund spaces are studied. Some characterizations for several kinds of generalized invexity and monotonicity concepts are given, using the properties of Mordukhovich limiting subdifferentials in Asplund spaces; and some applications in mathematical programming are provided. Also, some necessary and sufficient weak Pareto-optimality conditions for a multiple-objective optimization problem are proved.     

Generalized vector variational-like inequalities and vector optimization in Asplund spaces

Some properties of pseudoinvex functions, defined by means of limiting subdifferential, are obtained. Furthermore, the equivalence between vector variational-like inequalities involving limiting subdifferential and vector optimization problems are studied under pseudoinvexity condition.     

Asplund sets, differentiability and subdifferentiability of functions in Banach spaces

We show that Asplund sets are effective tools to study differentiability of Lipschitz functions, and ε-subdifferentiability of lower semicontinuous functions on general Banach spaces. If a locally Lipschitz function defined on an Asplund generated space has a minimal Clarke subdifferential mapping, then it is TBY-uniformly strictly differentiable on a dense Gδ subset of X. Examples are given of locally Lipschitz functions that are TBY-uniformly strictly differentiable everywhere, but nowhere Fréchet differentiable.     

Exact separation theorem for closed sets in Asplund spaces

In this paper, in terms of the Fréchet normal cone, we establish exact separation results for finitely many disjoint closed sets in an Asplund space, which supplement the extremal principle and some fuzzy separation theorems. As an application, we provide a new optimality condition for a constraint optimization problem in terms of Fréchet subdifferential and Fréchet normal cone.