Nonsmooth equations in optimization. Regularity, calculus, methods and applications.

*(English)*Zbl 1173.49300
Nonconvex Optimization and Its Applications 60. Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-0550-4/hbk). xxviii, 329 p. (2002).

The book deals with the theory of first and second order optimality conditions for nonlinear programming problems (NLP), as well as with the perturbation analysis of value function and solutions for such problems. The difficulty here lies in the fact that the cost function and constraints are not assumed to be twice continuously differentiable. The first chapters emphazise (strong) regularity of a multifunction, in the sense that the inverse mapping is locally (univalued) pseudo-Lipschitz. Regularity of the constraints, through an argument based on an exact penalty function, allows to obtain Lagrangre multipliers associated with local solutions. The book discusses various characterizations of regularity based on contingent and generalized derivatives, as well as Thibault’s limit set. The relations between these derivatives, chain rules and mean value theorems are presented. Another important parts of the book are the study of Kojima functions, which are a re-writing of optimality systems of NLP, assuming data to be \(C^{1,1}\) (they have Lipschitz derivatives), and the study on perturbation of NLP. The last chapters discuss Newton’s method for Lipschitz equations and their application to perturbed Kojima systems. Some results are presented in a Banach space setting, and some others such as the one on perturbed NLP in a finite dimensional setting. Probably due to space limitation, no link is discussed with semi infinite programming, which can be reformulated as a nonsmooth NLP. I found the book quite useful. The literature on this subject is, as the subject itself, quite technical and not easy to read. This book is a very convenient guide to a selection of the most important properties, in a field where there are still many discovers to be done.

Reviewer: Joseph Frédéric Bonnans (Le Chesnay)

##### MSC:

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

49J40 | Variational inequalities |

49M15 | Newton-type methods |

90C31 | Sensitivity, stability, parametric optimization |