Boris S. Mordukhovich

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 Frechet 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 di erential 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.

Complete characterization of openness, Metric regularity, And lipschitzian properties of multifunctions

We consider some basic properties of nonsmooth and set-valued mappings (multifunctions) connected with open and inverse mapping principles, distance estimates to the level sets (metric regularity), and a locally Lipschitzian behavior. These properties have many important applications to various problems in nonlinear analysis, optimization, control theory, etc., especially for studying sensitivity and stability questions with respect to perturbations of initial data and parameters. We establish interrelations between these properties and prove effective criteria for their fulfillment stated in terms of robust generalized derivatives for multifunctions and nonsmooth mappings

Relative Pareto minimizers for multiobjective problems: existence and optimality conditions

In this paper we introduce and study enhanced notions of relative Pareto minimizers for constrained multiobjective problems that are defined via several kinds of relative interiors of ordering cones and occupy intermediate positions between the classical notions of Pareto and weak Pareto efficiency/minimality. Using advanced tools of variational analysis and generalized differentiation, we establish the existence of relative Pareto minimizers for general multiobjective problems under a refined version of the subdifferential Palais-Smale condition for set-valued mappings with values in partially ordered spaces and then derive necessary optimality conditions for these minimizers (as well as for conventional efficient and weak efficient counterparts) that are new in both finite-dimensional and infinite-dimensional settings. Our proofs are based on variational and extremal principles of variational analysis; in particular, on new versions of the Ekeland variational principle and the subdifferential variational principle for set-valued and single-valued mappings in infinite-dimensional spaces.

Discrete Approximations and Refined Euler--Lagrange Conditions forNonconvex Differential Inclusions

This paper deals with the Bolza problem

Stability theory for parametric generalized equations and variational inequalities via nonsmooth analysis

(P)

Subgradients of marginal functions in parametric mathematical programming

for differential inclusions subject to general endpoint constraints. We pursue a twofold goal. First, we develop a finite difference method for studying

Coderivative Analysis of Quasi-variational Inequalities with Applications to Stability and Optimization

(P)

On Second-Order Subdifferentials and Their Applications

and construct a discrete approximation to

Stablity of Set-Valued Mappings In Infinite Dimensions: Point Criteria and Applications

(P)

Subdifferentials of value functions and optimality conditions for DC and bilevel infinite and semi-infinite programs

that ensures a strong convergence of optimal solutions. Second, we use this direct method to obtain necessary optimality conditions in a refined Euler--Lagrange form without standard convexity assumptions. In general, we prove necessary conditions for the so-called intermediate relaxed local minimum that takes an intermediate place between the classical concepts of strong and weak minima. In the case of a Mayer cost functional or boundary solutions to differential inclusions, this Euler--Lagrange form holds without any relaxation. The results obtained are expressed in terms of nonconvex-valued generalized differentiation constructions for nonsmooth mappings and sets.