Nonsmooth Equations in Optimization

1. Front Matter

   Pages i-xxviii

   PDF

2. Basic Concepts

   Pages 1-17

3. Regularity and Consequences

   Pages 19-60

4. Characterizations of Regularity by Derivatives

   Pages 61-70

5. Nonlinear Variations and Implicit Functions

   Pages 71-87

6. Closed Mappings in Finite Dimension

   Pages 89-103

7. Analysis of Generalized Derivatives

   Pages 105-147

8. Critical Points and Generalized Kojima-functions

   Pages 149-181

9. Parametric Optimization Problems

   Pages 183-230

10. Derivatives and Regularity of Further Nonsmooth Maps

    Pages 231-256

11. Newton’s Method for Lipschitz Equations

    Pages 257-273

12. Particular Newton Realizations and Solution Methods

    Pages 275-285

13. Basic Examples and Exercises

    Pages 287-301

14. Back Matter

    Pages 303-332

    PDF

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.

• Approximation
• Newton's method
• applied mathematics
• optimization

• Institute for Operations Research and Mathematical Methods of Economics, University of Zurich, Switzerland

  Diethard Klatte

• Institute of Mathematics, Faculty of Mathematics and Natural Sciences II, Humboldt University Berlin, Germany

  Bernd Kummer

Policies and ethics