“Multivalued Analysis” is the theory of set-valued maps (called multifonctions) and has important applications in many different areas. Multivalued analysis is a remarkable mixture of many different parts of mathematics such as point-set topology, measure theory and nonlinear functional analysis. It is also closely related to “Nonsmooth Analysis” (Chapter 5) and in fact one of the main motivations behind the development of the theory, was in order to provide necessary analytical tools for the study of problems in nonsmooth analysis. It is not a coincidence that the development of the two fields coincide chronologically and follow parallel paths. Today multivalued analysis is a mature mathematical field with its own methods, techniques and applications that range from social and economic sciences to biological sciences and engineering. There is no doubt that a modern treatise on “Nonlinear Functional Analysis” can not afford the luxury of ignoring multivalued analysis. The omission of the theory of multifunctions will drastically limit the possible applications.

SET-Valued Analysis

The paper is devoted to an analysis of a new constraint qualification and a derivation of the strongest existing optimality conditions for nonsmooth mathematical programming problems with equality and inequality constraints in terms of Demyanov-Rubinov-Polyakova quasidifferentials under the minimal possible assumptions. To this end, we obtain a novel description of convex subcones of the contingent cone to a set defined by quasidifferentiable equality and inequality constraints with the use of a new constraint qualification. We utilize these description and constraint qualification to derive the strongest existing optimality conditions for nonsmooth mathematical programming problems in terms of quasidifferentials under less restrictive assumptions than in previous studies. The main feature of the new constraint qualification and related optimality conditions is the fact that they depend on individual elements of quasidifferentials of the objective function and constraints and are not invariant with respect to the choise of quasidifferentials. To illustrate the theoretical results, we present two simple examples in which optimality conditions in terms of various subdifferentials (in fact, any outer semicontinuous/limiting subdifferential) are satisfied at a nonoptimal point, while the optimality conditions obtained in this paper do not hold true at this point, that is, optimality conditions in terms of quasidifferentials, unlike the ones in terms of subdifferentials, detect the nonoptimality of this point.

A New Constraint Qualification and Sharp Optimality Conditions for Nonsmooth Mathematical Programming Problems in Terms of Quasidifferentials

In this article we utilise abstract convexity theory in order to unify and generalize many different concepts from nonsmooth analysis. We introduce the concepts of abstract codifferentiability, abstract quasidifferentiability and abstract convex (concave) approximations of a nonsmooth function mapping a topological vector space to an order complete topological vector lattice. We study basic properties of these notions, construct elaborate calculus of abstract codifferentiable functions and discuss continuity of abstract codifferential. We demonstrate that many classical concepts of nonsmooth analysis, such as subdifferentiability and quasidifferentiability, are particular cases of the concepts of abstract codifferentiability and abstract quasidifferentiability. We also show that abstract convex and abstract concave approximations are a very convenient tool for the study of nonsmooth extremum problems. We use these approximations in order to obtain various necessary optimality conditions for nonsmooth nonconvex optimization problems with the abstract codifferentiable or abstract quasidifferentiable objective function and constraints. Then we demonstrate how these conditions can be transformed into simpler and more constructive conditions in some particular cases.

Abstract convex approximations of nonsmooth functions

This paper is devoted to an analysis of optimality conditions for nonsmooth mathematical programming problems with equality and inequality constraints in terms of Demyanov-Rubinov-Polyakova quasidifferentials. To this end, we obtain a novel description of convex subcones of the contingent cone to a set defined by quasidifferentiable equality and inequality constraints. With the use of this description we derive optimality conditions for nonsmooth mathematical programming problems in terms of quasidifferentials based on a new constraint qualification. The main feature of these optimality conditions and constraint qualification is the fact that they depend on individual elements of quasidifferentials of constraints. To illustrate the theoretical results, we present two simple examples in which the optimality conditions obtained in this paper are not satisfied at a given point, while optimality conditions in terms of various subdifferentials (in fact, any outer semicontinuous/limiting subdifferential) fail to disqualify this point as nonoptimal.

Sharp Optimality Conditions for Nonsmooth Mathematical Programming Problems in Terms of Quasidifferentials

The paper is devoted to the classical variational problem with a nonsmooth integrand of the functional to be minimized. The integrand is supposed to be subdifferentiable. Under some natural conditions the subdifferentiability of the functional considered is proved. The problem of finding the subdifferential descent direction is solved and the subdifferential descent method is applied to solve the original problem. The algorithm developed is demonstrated by examples. 

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The subdifferential descent method in a nonsmooth variational problem

The paper is devoted to an analysis of optimality conditions for nonsmooth multidimensional problems of the calculus of variations with various types of constraints, such as additional constraints at the boundary, isoperimetric constraints, and nonholonomic inequality constraints. To derive optimality conditions, we study generalised concepts of differentiability of nonsmooth functions called codifferentiability and quasidifferentiability. Under some natural and easily verifiable assumptions we prove that a nonsmooth integral functional defined on the Sobolev space is continuously codifferentiable and compute its codifferential and quasidifferential. Then we apply general optimality conditions for nonsmooth optimisation problems in Banach spaces to obtain optimality conditions for nonsmooth problems of the calculus of variations. Through a series of simple examples we demonstrate that our optimality conditions are sometimes better than existing ones in terms of various subdifferentials, in the sense that our optimality conditions can detect the non-optimality of a given point when subdifferential-based optimality conditions fail to disqualify this point as non-optimal. Apart from standard optimality conditions, we also study so-called inner variations of nonsmooth integral functionals and utilise them to extend the Noether equations for variational problems to a nonsmooth setting. With the use of these equations we obtain a nonsmooth law of conservation of energy for autonomous nonsmooth variational problems. Finally, we discuss some difficulties one faces when trying to extend famous Noether's theorem on symmetries and conservation laws to a nonsmooth case.

M. V. Dolgopolik

Papers

Constrained Nonsmooth Problems of the Calculus of Variations and Nonsmooth Noether Equations