Optimal Control of Nonconvex Differential Inclusions

The paper 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 paper

Suboptimality Conditions for Mathematical Programs with Equilibrium Constraints

In this paper we study mathematical programs with equilibrium constraints (MPECs) described by generalized equations in the extended form 0 is an element of the set G(x,y) + Q(x,y), where both mappings G and Q are set-valued. Such models arise, in particular, from certain optimization-related problems governed by variational inequalities and first-order optimality conditions in nondifferentiable programming. We establish new weak and strong suboptimality conditions for the general MPEC problems under consideration in finite-dimensional and infinite-dimensional spaces that do not assume the existence of optimal solutions. This issue is particularly important for infinite-dimensional optimization problems, where the existence of optimal solutions requires quite restrictive assumptions. Our techriiques are mainly based on modern tools of variational analysis and generalized differentiation revolving around the fundamental extremal principle in variational analysis and its analytic counterpart known as the subdifferential variational principle

A subdifferential criterion for calmness of multifunctions

A criterion for the calmness of a class of multifunctions between finite-dimensional spaces is derived in terms of subdifferential concepts developed by Mordukhovich. The considered class comprises general constraint set mappings as they occur in optimization or mappings associated with a certain type of variational systems. The criterion for calmness is obtained as an appropriate reduction of Mordukhovich's well-known characterization of the stronger Aubin property (roughly spoken, one may pass to the boundaries of normal cones or subdifferentials when aiming at calmness)

Variational Analysis of Evolution Inclusions

The paper is devoted to optimization problems of the Bolza and Mayer types for evolution systems governed by nonconvex Lipschitzian differential inclusions in Banach spaces under endpoint constraints described by finitely many equalities and inequalities. with generally nonsmooth functions. We develop a variational analysis of such roblems mainly based on their discrete approximations and the usage of advanced tools of generalized differentiation satisfying comprehensive calculus rules in the framework of Asplund (and hence any reflexive Banach) spaces. In this way we establish extended results on stability of discrete approximations (with the strong W^1,2-convergence of optimal solutions under consistent perturbations of endpoint constraints) and derive necessary optimality conditions for nonconvex discrete-time and continuous-time systems in the refined Euler-Lagrange and Weierstrass-Pontryagin forms accompanied by the appropriate transversality inclusions. In contrast to the case of geometric endpoint constraints in infinite dimensions, the necessary optimality conditions obtained in this paper do not impose any nonempty interiority /finite codimension/normal compactness assumptions. The approach and results developed in the paper make a bridge between optimal control/dynamic optimization and constrained mathematical programming problems in infinite-dimensional spaces

On Hölder calmness of solution mappings in parametric equilibrium problems

We consider parametric equilibrium problems in metric spaces. Sufficient conditions for the Hölder calmness of solutions are established. We also study the Hölder well-posedness for equilibrium problems in metric spaces

On calmness of a class of multifunctions

The paper deals with calmness of a class of multifunctions in finite dimensions. Its first part is devoted to various calmness criteria which are derived in terms of coderivatives and subdifferentials. The second part demonstrates the importance of calmness in several areas of nonsmoooth analysis. In particular, we focus on nonsmooth calculus and solution stability in mathematical programming and in equilibrium problems. The derived conditions find a number of applications there

Error bounds and metric subregularity

Necessary and sufficient criteria for metric subregularity (or calmness) of set-valued mappings between general metric or Banach spaces are treated in the framework of the theory of error bounds for a special family of extended real-valued functions of two variables. A classification scheme for the general error bound and metric subregularity criteria is presented. The criteria are formulated in terms of several kinds of primal and subdifferential slopes

Error bounds and their application

Our aim in this paper is to present sufficient conditions for error bounds in terms of Frechet and limiting Frechet subdifferentials outside of Asplund spaces. This allows us to develop sufficient conditions in terms of the approximate subdifferential for systems of the form (푥, 푦) ∈ 퐶 × 퐷, 푔(푥, 푦, 푢) = 0, where 푔 takes values in an infinite dimensional space and 푢 plays the role of a parameter. This symmetric structure offers us the choice to impose condtions either on 퐶 or 퐷. We use these results to prove nonemptyness and weak-star compactness of Fritz-John and Karuch-Kuhn-Tucker multiplier sets, to establish Lipschitz continuity of the value function and to compute its subdifferential and finally to obtain results on local controllability in control problems of nonconvex unbounded differential inclusions

Subdifferentials of distance functions in Banach spaces.

Ng, Kwong Wing.Thesis (M.Phil.)--Chinese University of Hong Kong, 2010.Includes bibliographical references (p. 123-126).Abstracts in English and Chinese.Abstract --- p.iAcknowledgments --- p.iiiContents --- p.vIntroduction --- p.viiChapter 1 --- Preliminaries --- p.1Chapter 1.1 --- Basic Notations and Conventions --- p.1Chapter 1.2 --- Fundamental Results in Banach Space Theory and Variational Analysis --- p.4Chapter 1.3 --- Set-Valued Mappings --- p.6Chapter 1.4 --- Enlargements and Projections --- p.8Chapter 1.5 --- Subdifferentials --- p.11Chapter 1.6 --- Sets of Normals --- p.18Chapter 1.7 --- Coderivatives --- p.24Chapter 2 --- The Generalized Distance Function - Basic Estimates --- p.27Chapter 2.1 --- Elementary Properties of the Generalized Distance Function --- p.27Chapter 2.2 --- Frechet-Like Subdifferentials of the Generalized Distance Function --- p.32Chapter 2.3 --- Limiting and Singular Subdifferentials of the Generalized Distance - Function --- p.44Chapter 3 --- The Generalized Distance Function - Estimates via Intermediate Points --- p.73Chapter 3.1 --- Frechet-Like and Limiting Subdifferentials of the Generalized Dis- tance Function via Intermediate Points --- p.74Chapter 3.2 --- Frechet and Proximal Subdifferentials of the Generalized Distance Function via Intermediate Points --- p.90Chapter 4 --- The Marginal Function --- p.95Chapter 4.1 --- Singular Subdifferentials of the Marginal Function --- p.95Chapter 4.2 --- Singular Subdifferentials of the Generalized Marginal Function . . --- p.102Chapter 5 --- The Perturbed Distance Function --- p.107Chapter 5.1 --- Elementary Properties of the Perturbed Distance Function --- p.107Chapter 5.2 --- The Convex Case - Subdifferentials of the Perturbed Distance Function --- p.111Chapter 5.3 --- The Nonconvex Case - Frechet-Like and Proximal Subdifferentials of the Perturbed Distance Function --- p.113Bibliography --- p.12

Weak Sharp Minima on Riemannian Manifolds

This is the first paper dealing with the study of weak sharp minima for constrained optimization problems on Riemannian manifolds, which are important in many applications. We consider the notions of local weak sharp minima, boundedly weak sharp minima, and global weak sharp minima for such problems and obtain their complete characterizations in the case of convex problems on finite-dimensional Riemannian manifolds and their Hadamard counterparts. A number of the results obtained in this paper are also new for the case of conventional problems in linear spaces. Our methods involve appropriate tools of variational analysis and generalized differentiation on Riemannian and Hadamard manifolds developed and efficiently implemented in this paper