Nonsmooth Equations In Optimization: Regularity, Calculus, Methods, And Applications

Nonsmooth Equations in Optimization Nonconvex Optimization and Its Applications Volume 60 Managing Editor: Panos Pardalos Advisory Board: J.R. Birge Northwestern University, U.S.A. Ding-Zhu Du University of Minnesota, U.S.A. C. A. Floudas Princeton University, U.S.A. J. Mockus Lithuanian Academy of Sciences, Lithuania H. D. Sherali Virginia Polytechnic Institute and State University, U.S.A. G. Stavroulakis Technical University Braunschweig, Germany The titles published in this series are listed at the end of this volume. Nonsmooth Equations in Optimization Regularity, Calculus, Methods and Applications by Diethard Klatte Institute for Operations Research and Mathematical Methods of Economics, University of Zurich, Switzerland and Bernd Kummer Institute of Mathematics, Faculty of Mathematics and Natural Sciences II, Humboldt University Berlin, Germany KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW eBook ISBN: Print ISBN: 0-306-47616-9 1-4020-0550-4 ©2002 Kluwer Academic Publishers New York, Boston, Dordrecht, London, Moscow Print ©2002 Kluwer Academic Publishers Dordrecht All rights reserved No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Kluwer Online at: and Kluwer's eBookstore at: http://kluweronline.com http://ebooks.kluweronline.com Contents Introduction xi List of Results xix Basic Notation xxv 1 Basic Concepts 1.1 Formal Settings 1.2 Multifunctions and Derivatives 1.3 Particular Locally Lipschitz Functions and Related Definitions Generalized Jacobians of Locally Lipschitz Functions Pseudo-Smoothness and D°f Piecewise Functions NCP Functions 1.4 Definitions of Regularity Definitions of Lipschitz Properties Regularity Definitions Functions and Multifunctions 1.5 Related Definitions Types of Semicontinuity Metric, Pseudo-, Upper Regularity; Openness with Linear Rate Calmness and Upper Regularity at a Set 1.6 First Motivations Parametric Global Minimizers Parametric Local Minimizers Epi-Convergence 1 1 2 4 4 4 5 5 6 6 7 9 10 10 12 13 14 15 16 17 2 Regularity and Consequences 2.1 Upper Regularity at Points and Sets Characterization by Increasing Functions Optimality Conditions Linear Inequality Systems with Variable Matrix Application to Lagrange Multipliers Upper Regularity and Newton’s Method 19 19 19 25 28 30 31 v Contents vi 2.2 Pseudo-Regularity 2.2.1 The Family of Inverse Functions 2.2.2 Ekeland Points and Uniform Lower Semicontinuity 2.2.3 Special Multifunctions Level Sets of L.s.c. Functions Cone Constraints Lipschitz Operators with Images in Hilbert Spaces Necessary Optimality Conditions 2.2.4 Intersection Maps and Extension of MFCQ Intersection with a Quasi-Lipschitz Multifunction Special Cases Intersections with Hyperfaces 32 34 37 43 43 44 46 47 49 49 54 58 3 Characterizations of Regularity by Derivatives 3.1 Strong Regularity and Thibault’s Limit Sets 3.2 Upper Regularity and Contingent Derivatives 3.3 Pseudo-Regularity and Generalized Derivatives Contingent Derivatives Proper Mappings Closed Mappings Coderivatives Vertical Normals 61 61 63 63 64 64 64 66 67 4 Nonlinear Variations and Implicit Functions 4.1 Successive Approximation and Persistence of Pseudo-Regularity 4.2 Persistence of Upper Regularity Persistence Based on Kakutani’s Fixed Point Theorem Persistence Based on Growth Conditions 4.3 Implicit Functions 71 72 77 77 79 82 5 Closed Mappings in Finite Dimension 5.1 Closed Multifunctions in Finite Dimension 5.1.1 Summary of Regularity Conditions via Deriva