Constructive nonsmooth analysis. (English) Zbl 0887.49014

Approximation and Optimization. 7. Frankfurt/Main: Verlag Peter Lang. 416 p. (1995).
The present book is an extended and entirely revised update of two previous books by the same authors [“Quasidifferential calculus” (1986; Zbl 0712.49012), “Foundations of nonsmooth analysis and quasidifferential calculus” (1990; Zbl 0728.49001)]. The authors deal with the calculus of quasi- and codifferentiable functions and with its application to fundamental problems of nonsmooth analysis and optimization. In the first four chapters several aspects of approximations of nonsmooth functions, sets and mappings are considered, especially, Dini and Hadamard derivatives, several cone approximations, Clarke derivatives and subdifferentials, quasidifferentials, codifferentials. For each aspect of approximation, the main rules, the calculus and the characteristic properties are given. Some comparisons between several directions of approximations are outlined. Remarkable is, that the codifferentiability allows to consider kinds of higher order differentiability and approximation of nonsmooth functions similar to Taylor’s expansion. First applications are given to unconstrained and constrained (exact penalty) extremal problems (Chapter 5). Besides optimality conditions of first and second order basing on the above introduced generalized derivatives some possible numerical method called codifferential descent is suggested. Its convergence is proved. However, numerical examples are not given. The following sections are devoted to the construction of descent directions of several classes of nonsmooth functions. Especially, these sections contain detailed informations and ideas for further possible numerical usage of the outlined theory. In Chapter 6 the implicit function and inverse function problem is considered for Lipschitz functions and local Lipschitz mappings (Clarke) as well as for quasidifferentiable functions. A directionally implicit function theorem is derived under weaker conditions for the generalized Jacobian. Using the Ekeland variational principle inverse function theorems for multi-valued mappings are proved. Some basic things are shortly outlined in Appendix I (convex analysis), II (multi-valued mappings), III (quasidifferential calculus in Banach \(K\)-spaces), IV (quasilinear algebra), V (difference of sets, open questions). Short bibliographical notes and a broad and comprehensive bibliography with 300 references close the book.
For reading the book it is not necessary to be familiar with the foundations of nonsmooth analysis. The book is self-contained and well readable. The proofs are given in detail. The best one are the many examples and figures making the contents easily to comprehend. The book gives a review of the known and some latest results in quasidifferentiable programming and related fields. With respect to its structure it can be well used for lectures in advanced studies and as handbook for scientists in nonsmooth analysis.


49J52 Nonsmooth analysis
49-02 Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control