Nonsmooth Equations in Optimization: Regularity, Calculus, Methods and Applications (Nonconvex Optimization and Its Applications (60))

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. The book establishes links between regularity and derivative concepts of nonsmooth analysis and studies of solution methods and stability for optimization, complementarity and equilibrium problems. In developing necessary tools, it presents, in particular: - an extended analysis of Lipschitz functions and the calculus of their generalized derivatives, including regularity, successive approximation and implicit functions for multivalued mappings; - a unified theory of Lipschitzian critical points in optimization and other variational problems, with relations to reformulations by penalty, barrier and NCP functions; - an analysis of generalized Newton methods based on linear and nonlinear approximations; - the interpretation of hypotheses, generalized derivatives and solution methods in terms of original data and quadratic approximations; - a rich collection of instructive examples and exercises. Audience: Researchers, graduate students and practitioners in various fields of applied mathematics, engineering, OR and economics. Also university teachers and advanced students who wish to get insights into problems, future directions and recent developments. The book establishes links between regularity and derivative concepts of nonsmooth analysis and studies of solution methods and stability for optimization, complementarity and equilibrium problems. In developing necessary tools, it presents, in particular: an extended analysis of Lipschitz functions and the calculus of their generalized derivatives, including regularity, successive approximation and implicit functions for multivalued mappings; a unified theory of Lipschitzian critical points in optimization and other variational problems, with relations to reformulations by penalty, barrier and NCP functions; an analysis of generalized Newton methods based on linear and nonlinear approximations; the interpretation of hypotheses, generalized derivatives and solution methods in terms of original data and quadratic approximations; a rich collection of instructive examples and exercises.£/LIST£ Audience: Researchers, graduate students and practitioners in various fields of applied mathematics, engineering, OR and economics. Also university teachers and advanced students who wish to get insights into problems, future directions and recent developments.

investigates Optimization Problems, Solutions Of Generalized Equations, And Complementary Problems From The Perspective Of Nonsmooth Analysis. Klatte (university Of Zurich) And Kummer (humboldt University, Berlin) Present Conditions For Lipschitz Properties Of Multi-valued Maps And The Related Types Of Regularity, And Apply The Regularity Characteristics For Zeros Of Generalized Kojima Functions To Critical Points, Stationary Solutions, And Local Minimizers Of Parametric Nonlinear Optimization Problems In Finitely Many Variables. Annotation (c)2003 Book News, Inc., Portland, Or

"This book establishes links between regularity and derivative concepts of nonsmooth analysis and studies of solution methods and stability for optimization, complementarity and equilibrium problems." "Audience: Researchers, graduate students and practitioners in various fields of applied mathematics, engineering, OR and economics. Also university teachers and advanced students who wish to get insights into problems, future directions and recent developments."--Jacket