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Winfried Schirotzek Nonsmooth Analysis 31 With Figures WinfriedSchirotzek Institutfu¨rAnalysis FachrichtungMathematik TechnischeUniversita¨tDresden 01062Dresden Germany e-mail:[email protected] Mathematics Subject Classification (2000): 49-01, 49-02, 49J50, 49J52, 49J53, 49K27, 49N15,58C06,58C20,58E30,90C48 LibraryofCongressControlNumber:2007922937 ISBN-10:3-540-71332-8SpringerBerlinHeidelbergNewYork ISBN-13:978-3-540-71332-6SpringerBerlinHeidelbergNewYork Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthemater- ialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Dupli- cationofthispublicationorpartsthereofispermittedonlyundertheprovisionsoftheGerman CopyrightLawofSeptember9,1965,initscurrentversion,andpermissionforusemustalways beobtainedfromSpringer.ViolationsareliableforprosecutionundertheGermanCopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springer.com ◦ c Springer-VerlagBerlinHeidelberg2007 Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispub- licationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnames areexemptfromtherelevantprotectivelawsandregulationsandthereforefreefor generaluse. Coverdesign:WMXDesign,Heidelberg TypesettingbytheauthorsandSPiusingaSpringerLATEXmacropackage Printedonacid-freepaper SPIN:12029495 41/2141/SPi 543210 To the memory of my parents Preface One of the sources of the classical differential calculus is the search for minimum or maximum points of a real-valued function. Similarly, nonsmooth analysisoriginatesinextremumproblemswithnondifferentiabledata.Bynow, a broad spectrum of refined concepts and methods modeled on the theory of differentiation has been developed. Theideaunderlyingthepresentationofthematerialinthisbookistostart with simple problems treating them with simple methods, gradually passing to more difficult problems which need more sophisticated methods. In this sense, we pass from convex functionals via locally Lipschitz continuous functionals to general lower semicontinuous functionals. The book does not aim at being comprehensive but it presents a rather broad spectrum of important and applicable results of nonsmooth analysis in normed vector spaces. Each chapter ends with references to the literature and with various exercises. ThebookgrewoutofagraduatecoursethatIrepeatedlyheldattheTech- nische Universität Dresden. Susanne Walther and Konrad Groh, participants of one of the courses, pointed out misprints in an early script preceding the book.IamparticularlygratefultoHeidrunPu¨hlandHans-PeterSchefflerfor a time of prolific cooperation and to the latter also for permanent technical support.TheInstitutfu¨rAnalysisoftheTechnischeUniversitätDresdenpro- vided me with the facilities to write the book. I thank Quji J. Zhu for useful discussions and two anonymous referees for valuable suggestions. I gratefully acknowledge the kind cooperation of Springer, in particular the patient support by Stefanie Zoeller, as well as the careful work of Nandini Loganathan, project manager of Spi (India). My warmest thanks go to my wife for everything not mentioned above. Dresden, December 2006 Winfried Schirotzek Contents Introduction................................................... 1 1 Preliminaries .............................................. 5 1.1 Terminology............................................ 5 1.2 Convex Sets in Normed Vector Spaces ..................... 6 1.3 Convex Functionals: Definitions and Examples.............. 8 1.4 Continuity of Convex Functionals ......................... 11 1.5 Sandwich and Separation Theorems ....................... 13 1.6 Dual Pairs of Vector Spaces .............................. 19 1.7 Lower Semicontinuous Functionals ........................ 22 1.8 Bibliographical Notes and Exercises ....................... 24 2 The Conjugate of Convex Functionals ..................... 27 2.1 The Gamma Regularization .............................. 27 2.2 Conjugate Functionals................................... 29 2.3 A Theorem of Hörmander and the Bipolar Theorem......... 34 2.4 The Generalized Farkas Lemma........................... 36 2.5 Bibliographical Notes and Exercises ....................... 38 3 Classical Derivatives....................................... 39 3.1 Directional Derivatives .................................. 39 3.2 First-Order Derivatives .................................. 41 3.3 Mean Value Theorems................................... 44 3.4 Relationship between Differentiability Properties............ 46 3.5 Higher-Order Derivatives ................................ 48 3.6 Some Examples......................................... 49 3.7 Implicit Function Theorems and Related Results............ 51 3.8 Bibliographical Notes and Exercises ....................... 57 X Contents 4 The Subdifferential of Convex Functionals ................. 59 4.1 Definition and First Properties ........................... 59 4.2 Multifunctions: First Properties........................... 63 4.3 Subdifferentials, Fréchet Derivatives, and Asplund Spaces .... 64 4.4 Subdifferentials and Conjugate Functionals................. 73 4.5 Further Calculus Rules .................................. 76 4.6 The Subdifferential of the Norm .......................... 78 4.7 Differentiable Norms .................................... 83 4.8 Bibliographical Notes and Exercises ....................... 89 5 Optimality Conditions for Convex Problems ............... 91 5.1 Basic Optimality Conditions.............................. 91 5.2 Optimality Under Functional Constraints .................. 92 5.3 Application to Approximation Theory ..................... 96 5.4 Existence of Minimum Points and the Ritz Method.......... 99 5.5 Application to Boundary Value Problems ..................105 5.6 Bibliographical Notes and Exercises .......................110 6 Duality of Convex Problems ...............................111 6.1 Duality in Terms of a Lagrange Function...................111 6.2 Lagrange Duality and Gâteaux Differentiable Functionals ....116 6.3 Duality of Boundary Value Problems ......................118 6.4 Duality in Terms of Conjugate Functions...................122 6.5 Bibliographical Notes and Exercises .......................129 7 Derivatives and Subdifferentials of Lipschitz Functionals...131 7.1 Preview: Derivatives and Approximating Cones .............131 7.2 Upper Convex Approximations and Locally Convex Functionals ..........................135 7.3 The Subdifferentials of Clarke and Michel–Penot............139 7.4 Subdifferential Calculus..................................146 7.5 Bibliographical Notes and Exercises .......................153 8 Variational Principles......................................155 8.1 Introduction............................................155 8.2 The Loewen–Wang Variational Principle ...................156 8.3 The Borwein–Preiss Variational Principle ..................161 8.4 The Deville–Godefroy–Zizler Variational Principle...........162 8.5 Bibliographical Notes and Exercises .......................166 9 Subdifferentials of Lower Semicontinuous Functionals......167 9.1 Fréchet Subdifferentials: First Properties...................167 9.2 Approximate Sum and Chain Rules .......................172 9.3 Application to Hamilton–Jacobi Equations .................181 9.4 An Approximate Mean Value Theorem ....................182 9.5 Fréchet Subdifferential vs. Clarke Subdifferential ............184 Contents XI 9.6 Multidirectional Mean Value Theorems ....................185 9.7 The Fréchet Subdifferential of Marginal Functions...........190 9.8 Bibliographical Notes and Exercises .......................193 10 Multifunctions.............................................195 10.1 The Generalized Open Mapping Theorem ..................195 10.2 Systems of Convex Inequalities ...........................197 10.3 Metric Regularity and Linear Openness....................200 10.4 Openness Bounds of Multifunctions .......................209 10.5 Weak Metric Regularity and Pseudo-Lipschitz Continuity ....211 10.6 Linear Semiopenness and Related Properties ...............213 10.7 Linearly Semiopen Processes .............................217 10.8 Maximal Monotone Multifunctions ........................219 10.9 Convergence of Sets .....................................225 10.10 Bibliographical Notes and Exercises .......................227 11 Tangent and Normal Cones................................231 11.1 Tangent Cones: First Properties ..........................231 11.2 Normal Cones: First Properties ...........................237 11.3 Tangent and Normal Cones to Epigraphs ..................241 11.4 Representation of Tangent Cones .........................245 11.5 Contingent Derivatives and a Lyusternik Type Theorem .....252 11.6 Representation of Normal Cones ..........................255 11.7 Bibliographical Notes and Exercises .......................261 12 Optimality Conditions for Nonconvex Problems ...........265 12.1 Basic Optimality Conditions..............................265 12.2 Application to the Calculus of Variations ..................267 12.3 Multiplier Rules Involving Upper Convex Approximations....272 12.4 Clarke’s Multiplier Rule .................................278 12.5 Approximate Multiplier Rules ............................280 12.6 Bibliographical Notes and Exercises .......................283 13 Extremal Principles and More Normals and Subdifferentials .......................................285 13.1 Mordukhovich Normals and Subdifferentials ................285 13.2 Coderivatives...........................................294 13.3 Extremal Principles Involving Translations .................301 13.4 Sequentially Normally Compact Sets ......................309 13.5 Calculus for Mordukhovich Subdifferentials.................315 13.6 Calculus for Mordukhovich Normals.......................320 13.7 Optimality Conditions...................................323 13.8 The Mordukhovich Subdifferential of Marginal Functions.....327 13.9 A Nonsmooth Implicit Function Theorem ..................330 13.10 An Implicit Multifunction Theorem .......................334 XII Contents 13.11 An Extremal Principle Involving Deformations..............337 13.12 Application to Multiobjective Optimization ................340 13.13 Bibliographical Notes and Exercises .......................343 Appendix: Further Topics .....................................347 References.....................................................351 Notation.......................................................363 Index..........................................................366 Introduction Minimizing or maximizing a function subject to certain constraints is one of the most important problems of real life and consequently of mathematics. Among others, it was this problem that stimulated the development of differential calculus. Given a real-valued function f on a real normed vector space E and a nonempty subset A of E, consider the following problem: (Min)Minimize f(x) subject to x∈A. Let x¯ be a local solution of (Min). If x¯ is an interior point of A and f is differentiable (in some sense) at x¯, then x¯ satisfies the famous Fermat rule (cid:1) f (x¯)=o, which is thus a necessary optimality condition. If x¯ ∈ A is not an interior pointofAbutf goesontobedifferentiableatx¯,thenanecessaryoptimality condition still holds as a variational inequality, which for A convex reads (cid:2)f(cid:1)(x¯),x−x¯(cid:3)≥0 for any x∈A. (0.1) The assumption that f be differentiable at x¯ is not intrinsic to problem (Min). Consider, for example, the classical problem of Chebyshev approximation, which is (Min) with E := C[a,b], the normed vector space of all continuous functions x:[a,b]→R, and f(x):=(cid:6)x−z(cid:6)∞ := max |x(t)−z(t)|, a≤t≤b wherez ∈C[a,b]\Aisgiven.Inthiscasethefunctionalf failstobe(Gâteaux) differentiable at “most” points x ∈ C[a,b] and so the above-mentioned app- roach no longer works. However, if f is aconvex functional, as is the functional in theChebyshev approximation problem, then a useful substitute for a nonexisting derivative