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.     

On Metric and Calmness Qualification Conditions in Subdifferential Calculus

The paper contains two groups of results. The first are criteria for calmness/subregularity for set-valued mappings between finite-dimensional spaces. We give a new sufficient condition whose subregularity part has the same form as the coderivative criterion for “full” metric regularity but involves a different type of coderivative which is introduced in the paper. We also show that the condition is necessary for mappings with convex graphs. The second group of results deals with the basic calculus rules of nonsmooth subdifferential calculus. For each of the rules we state two qualification conditions: one in terms of calmness/subregularity of certain set-valued mappings and the other as a metric estimate (not necessarily directly associated with aforementioned calmness/subregularity property). The conditions are shown to be weaker than the standard Mordukhovich–Rockafellar subdifferential qualification condition; in particular they cover the cases of convex polyhedral set-valued mappings and, more generally, mappings with semi-linear graphs. Relative strength of the conditions is thoroughly analyzed. We also show, for each of the calculus rules, that the standard qualification conditions are equivalent to “full” metric regularity of precisely the same mappings that are involved in the subregularity version of our calmness/subregularity condition. The research of Jiří V. Outrata was supported by the grant A 107 5402 of the Grant Agency of the Academy of Sciences of the Czech Republic.     

Coderivative Analysis of Variational Systems

The paper mostly concerns applications of the generalized differentiation theory in variational analysis to Lipschitzian stability and metric regularity of variational systems in infinite-dimensional spaces. The main tools of our analysis involve coderivatives of set-valued mappings that turn out to be proper extensions of the adjoint derivative operator to nonsmooth and set-valued mappings. The involved coderivatives allow us to give complete dual characterizations of certain fundamental properties in variational analysis and optimization related to Lipschitzian stability and metric regularity. Based on these characterizations and extended coderivative calculus, we obtain efficient conditions for Lipschitzian stability of variational systems governed by parametric generalized equations and their specifications.     

Viscosity Coderivatives and Their Limiting Behavior in Smooth Banach Spaces

This paper concerns with generalized differentiation of set-valued and nonsmooth mappings between Banach spaces. We study the so-called viscosity coderivatives of multifunctions and their limiting behavior under certain geometric assumptions on spaces in question related to the existence of smooth bump functions of any kind. The main results include various calculus rules for viscosity coderivatives and their topological limits. They are important in applications to variational analysis and optimization.     

Calculus of directional coderivatives and normal cones in Asplund spaces

We study the directional Mordukhovich normal cones to nonsmooth sets, coderivatives of set-valued mappings in Asplund spaces and establish extensive calculus results on these constructions under various operations of sets and mappings. We also develop calculus of the directional sequential normal compactness both in general Banach spaces and in Asplund spaces.     

集值映射的次微分的性质及其应用(英)

本文讨论了文章``Subgradient of S-convex set-valued mappings and weak efficient solutions"(Appl.Math. J. Chinese Univ. 1998, 13(4): 463-472) 中引入的集值映射的次微分的性质及应用.利用相依导数的性质,讨论次微分的性质,并得到了两个集值映射的和、复合以及交的次微分的运算法则.最后,通过这种次微分得到了集值优化问题最优性条件的充要条件,同时推广了此文中的定理7.     

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.

     

一个精细的Bishop-Phelps定理

讨论了凸函数的次微分映射和凸集的支撑点集之间的内在关系，由此本给出了由凸函数的次微分映射所刻划的一个精细的Bishop-Phelps定理．     

Metric regularity, tangential distances and generalized differentiation in Banach spaces

In this paper we establish new generalized differentiation rules in general Banach spaces regarding normal cones to set images under functions, coderivatives of compositions of set-valued mappings, as well as calculus results for normal compactness of sets and their images. In addition to the metric regularity of mappings, our results involve tangential distances of sets for which we also provide a fairly complete study by exploring its variations, basic properties, as well as relations to similar notions. Some related results are also established.     

Fixed Points and Zeros for Set Valued Mappings on Riemannian Manifolds: A Subdifferential Approach

In this paper we establish several results which allow to find fixed points and zeros of set-valued mappings on Riemannian manifolds. In order to prove these results we make use of subdifferential calculus. We also give some useful applications.     

Subgradient of distance functions with applications to Lipschitzian stability

The paper is devoted to studying generalized differential properties of distance functions that play a remarkable role in variational analysis, optimization, and their applications. The main object under consideration is the distance function of two variables in Banach spaces that signifies the distance from a point to a moving set. We derive various relationships between Fréchet-type subgradients and limiting (basic and singular) subgradients of this distance function and corresponding generalized normals to sets and coderivatives of set-valued mappings. These relationships are essentially different depending on whether or not the reference point belongs to the graph of the involved set-valued mapping. Our major results are new even for subdifferentiation of the standard distance function signifying the distance between a point and a fixed set in finite-dimensional spaces. The subdifferential results obtained are applied to deriving efficient dual-space conditions for the local Lipschitz continuity of distance functions generated by set-valued mappings, in particular, by those arising in parametric constrained optimization. Dedicated to Terry Rockafellar in honor of his 70th birthday. This research was partially supported by the National Science Foundation under grant DMS-0304989 and by the Australian Research Council under grant DP-0451158.     

法向锥与集值映射的单调极大性

研究局部凸空间凸集的法向锥及单调映射与下半连续凸函数的次微分之和的单调极大性．并讨论集值映射循环单调的极大性．     

Generalized penalization and maximization of vectorial nonsmooth functions

We use directional Lipschitz concepts and a minimal time function with respect to a set of directions in order to derive generalized penalization results for Pareto minimality in set-valued constrained optimization. Then, we obtain necessary optimality conditions for maximization in constrained vector optimization in terms of generalized differentiation objects. To the latter aim, we deduce first some enhanced calculus rules for coderivatives of the difference of two mappings. All the main results of this paper are tailored to model directional features of the optimization problem under study.     

广义扰动映射的上导数

主要讨论了两集值映射和的上导数．在比标准约束品性弱的条件下得到了两个集值映射和的上导数与两集值映射上导数的和之间的包含关系,并将此结论用于讨论广义扰动映射的上导数,得到广义扰动映射的上导数的上界估计．     

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.     

Weak Subdifferentials for Set-Valued Mappings

The purpose of this paper is to study the weak subdifferential for set-valued mappings, which was introduced by Chen and Jahn (Math. Methods Oper. Res., 48:187–200, 1998). Two existence theorems of weak subgradients for set-valued mappings are obtained. Moreover, some properties of the weak subdifferential for set-valued mappings are derived. Our results improve the corresponding ones in the literature. Some examples are given to illustrate our results.     

Set-valued mapping monotonicity as characterization of D.C functions

In this paper, using a result of Kung-Ching Chang [6], we give a characterization of locally Lipschitz functions which are differences of convex functions defined on a Banach space (not necessarily Asplund) in terms on maximal cyclically monotone set-valued mappings. A subdifferential integration of locally D.C functions. is also given.     

扰动集值优化问题超有效点集的次微分稳定性

在锥序Banach向量空间引入了集值映射在超有效意下的次微分(次梯度);在一定的条件下,证明次微分(次梯度)的存在性;得到了序扰动、双扰动集值优化问题超有效点集在次微分意义下的稳定性．     

Stability of Saddle Points Via Explicit Coderivatives of Pointwise Subdifferentials

We derive stability criteria for saddle points of a class of nonsmooth optimization problems in Hilbert spaces arising in PDE-constrained optimization, using metric regularity of infinite-dimensional set-valued mappings. A main ingredient is an explicit pointwise characterization of the regular coderivative of the subdifferential of convex integral functionals. This is applied to several stability properties for parameter identification problems for an elliptic partial differential equation with non-differentiable data fitting terms.     

Lipschitz properties of nonsmooth functions and set-valued mappings via generalized differentiation

In this paper, we revisit the Mordukhovich subdifferential criterion for Lipschitz continuity of nonsmooth functions and the coderivative criterion for the Aubin/Lipschitz-like property of set-valued mappings in finite dimensions. The criteria are useful and beautiful results in modern variational analysis showing the state of the art of the field. As an application, we establish necessary and sufficient conditions for Lipschitz continuity of the minimal time function and the scalarization function, which play an important role in many aspects of nonsmooth analysis and optimization.