Bingwu Wang

Necessary Suboptimality and Optimality Conditions via Variational Principles

The paper aims to develop some basic principles and tools of nonconvex variational analysis with applications to necessary suboptimality and optimality conditions for constrained optimization problems in infinite dimensions. We establish a certain subdifferential variational principle as a new characterization of Asplund spaces. This result is different from conventional support forms of variational principles and appears to be convenient for applications to nonsmooth optimization. Based on the subdifferential variational principle, we obtain new necessary conditions for suboptimal solutions in general nonsmooth optimization problems with equality, inequality, and set constraints in Asplund spaces. In this way we establish the so-called sequential normal compactness properties of constraint sets that play an essential role in infinite-dimensional variational analysis and its applications. As a by-product of our approach, we derive various forms of necessary optimality conditions for nonsmooth constrained problems in infinite dimensions, which extend known results in that direction.

Restrictive metric regularity and generalized differential calculus in Banach spaces

We consider nonlinear mappings f:X→Y between Banach spaces and study the notion of restrictive metric regularity of f around some point x¯, that is, metric regularity of f from X into the metric space E=f(X). Some sufficient as well as necessary and sufficient conditions for restrictive metric regularity are obtained, which particularly include an extension of the classical Lyusternik-Graves theorem in the case when f is strictly differentiable at x¯ but its strict derivative ∇f(x¯) is not surjective. We develop applications of the results obtained and some other techniques in variational analysis to generalized differential calculus involving normal cones to nonsmooth and nonconvex sets, coderivatives of set-valued mappings, as well as first-order and second-order subdifferentials of extended real-valued functions.

Calculus of sequential normal compactness in variational analysis

Abstract In this paper we study some properties of sets, set-valued mappings, and extended-real-valued functions unified under the name of “sequential normal compactness.” These properties automatically hold in finite-dimensional spaces, while they play a major role in infinite-dimensional variational analysis. In particular, they are essential for calculus rules involving generalized differential constructions, for stability and metric regularity results and their broad applications, for necessary optimality conditions in constrained optimization and optimal control, etc. This paper contains principal results ensuring the preservation of sequential normal compactness properties under various operations over sets, set-valued mappings, and functions.

Extensions of generalized differential calculus in asplund spaces

Abstract 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.

Generalized differentiation of parameter-dependent sets and mappings

This article concerns new aspects of generalized differentiation theory that plays a crucial role in many areas of modern variational analysis, optimization and their applications. In contrast to the majority of previous developments, we focus here on generalized differentiation of parameter-dependent objects (sets, set-valued mappings and nonsmooth functions), which naturally appear, e.g. in parametric optimization and related topics. The basic generalized differential constructions needed in this case are different for those known in parameter-independent settings, while they still enjoy comprehensive calculus rules developed in this article. †Dedicated to Bert Jongen in honor of his 60th birthday.

Differentiability and regularity of Lipschitzian mappings

We introduce new differentiability properties of functions between Banach spaces and establish their relationships with graphical regularity of Lipschitzian single-valued and set-valued mappings. The proofs are based on advanced tools of nonsmooth variational analysis including new results on coderivative scalarization and normal cone calculus.

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.

Outer semicontinuity of subdifferential and normal cone mappings

In finite dimensions, the outer semicontinuity of a set-valued mapping is equivalent to the closedness of its graph. In this article, we study the outer semicontinuity of set-valued mappings in connection with their convexifications and linearizations in finite and infinite dimensions. The results are specified to the case where the mappings involved are given by subdifferentials of extended real-valued functions or normal cones to sets. Our developments are important for applications to second-order calculus in variational analysis in which the outer semicontinuity plays a crucial role.

Necessary optimality and suboptimality conditions for nonsmooth problems

The paper aims to develop some basic tools of nonconvex variational analysis with applications to necessary suboptimality and optimality conditions for constrained optimization problems in infinite dimensions. We establish a certain subdifferential variational principle as a new characterization of Asplund spaces and reveal the so-called sequential normal compactness properties of constraint sets that play an essential role in an infinite-dimensional variational analysis and its applications. In light of these tools we obtain new necessary conditions for suboptimal and optimal solutions in general nonsmooth optimization problems with equality, inequality, and set constraints in Asplund spaces.