Nonsmooth Analysis: Differential Calculus of Nondifferentiable Mappings

Abstract

A new approach to local analysis of nonsmooth mappings from one Banach space into another is suggested. The approach is essentially based on the use of set-valued mappings of a special kind, called fans, for local approximation. Convex sets of linear operators provide an example of fans. Generally, fans can be considered a natural set-valued extension of linear operators. The first part of the paper presents a study of fans; the second is devoted to calculus and includes extensions of the main theorems of classical calculus.

Nonsmooth sequential analysis in Asplund spaces

“…Note that constructions (9.1) and (9.3) coincide with those introduced in [20] under the name of M -subdifferential and M -normal cone, respectively.…”

“…Moreover, one can check directly from the definitions that any locally Lipschitzian mapping acting between Banach spaces is strictly Lipschitzian atx if it has a norm-compact-valued "strict prederivative" in the sense of Ioffe [20]. Thus the class of strictly Lipschitzian mappings covers all strictly differentiable mappings and all compositions H • F of a Lipschitz continuous mapping F with a strictly differentiable mapping H whose derivative is a compact operator.…”

“…On the other line of development, a series of infinite dimensional extensions of the nonconvex constructions in [41,42] were introduced and studied by Ioffe [20,23,24] on the basis of topological limits of Dini subdifferentials and ε-subdifferentials. Such constructions, called "approximate subdifferentials", are well defined in more general spaces, but all of them (including their "nuclei") may be broader than the Kruger-Mordukhovich extension even for Lipschitz functions on Banach spaces with Fréchet differentiable renorms; see Section 9. This paper is devoted to the development of sequential subdifferential and related constructions in infinite dimensions, mostly for the class of Banach spaces called Asplund spaces.…”

Nonsmooth analysis and optimization on partially ordered vector spaces

“…Since [18]. A real-valued function defined on a topological vector space E is quasidifferentiable at e E in the [29] (where such multifunctions are called fans) and lead to necessary conditions which have as special cases the necessary conditions of Clarke [24], Hiriart-Urruty [I, 38,39] and Ioffe [40]. Ioffe [30] has in fact used the concept of fan to develop more general necessary conditions.…”

“…For excellent summaries of the theory, motivation and applications of generalized gradients and extensive references we refer the reader to Clarke [2], Hiriart-Urruty [1] and Rockafellar [26]; in addition, Borwein and Strojwas [27] provide an insightful comparison of several recent directional derivatives and generalized gradients of the same genre as Clarke's gradient. The excellent papers by Papageorgiou [17,28] and Ioffe [29,30] provide many fundamental results in nonsmooth analysis for vector-valued mappings.…”