Nonsmooth Equations in Optimization - B. Kummer

. .

Résumé :
Many questions dealing with solvability, stability and solution methods for va- ational inequalities or equilibrium, optimization and complementarity problems lead to the analysis of certain (perturbed) equations. This often requires a - formulation of the initial model being under consideration. Due to the specific of the original problem, the resulting equation is usually either not differ- tiable (even if the data of the original model are smooth), or it does not satisfy the assumptions of the classical implicit function theorem. This phenomenon is the main reason why a considerable analytical inst- ment dealing with generalized equations (i.e., with finding zeros of multivalued mappings) and nonsmooth equations (i.e., the defining functions are not c- tinuously differentiable) has been developed during the last 20 years, and that under very different viewpoints and assumptions. In this theory, the classical hypotheses of convex analysis, in particular, monotonicity and convexity, have been weakened or dropped, and the scope of possible applications seems to be quite large. Briefly, this discipline is often called nonsmooth analysis, sometimes also variational analysis. Our book fits into this discipline, however, our main intention is to develop the analytical theory in close connection with the needs of applications in optimization and related subjects. Main Topics of the Book 1. Extended analysis of Lipschitz functions and their generalized derivatives, including ?Newton maps? and regularity of multivalued mappings. 2. Principle of successive approximation under metric regularity and its - plication to implicit functions.

Sommaire:
Basic Concepts.- Regularity and Consequences.- Characterizations of Regularity by Derivatives.- Nonlinear Variations and Implicit Functions.- Closed Mappings in Finite Dimension.- Analysis of Generalized Derivatives.- Critical Points and Generalized Kojima-functions.- Parametric Optimization Problems.- Derivatives and Regularity of Further Nonsmooth Maps.- Newton's Method for Lipschitz Equations.- Particular Newton Realizations and Solution Methods.- Basic Examples and Exercises.