One of the sources of the classical di?erential calculus is the search for m- imum or maximum points of a real-valued function. Similarly, nonsmooth analysisoriginatesinextremumproblemswithnondi?erentiabledata.Bynow, a broad spectrum of re?ned concepts and methods modeled on the theory of di?erentiation has been developed. Theideaunderlyingthepresentationofthematerialinthisbookistostart with simple problems treating them with simple methods, gradually passing to more di?cult problems which need more sophisticated methods. In this sense, we pass from convex functionals via locally Lipschitz continuous fu- tionals 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. ThebookgrewoutofagraduatecoursethatIrepeatedlyheldattheTe- nische Universitšt Dresden. Susanne Walther and Konrad Groh, participants of one of the courses, pointed out misprints in an early script preceding the book. I am particularly grateful to Heidrun Pu ̈hl and Hans-Peter Sche?er for a time of proli?c cooperation and to the latter also for permanent technical support. The Institut fu ̈r Analysis of the Technische Universit ̈ atDresdenp- 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 s- port 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.
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The Conjugate of Convex Functionals
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analogous Asplund space assertion Assume assumptions Borwein bounded calculus chain rule closed subset compact conclude continuously differentiable convergent convex cone convex functional convex set convex subset Corollary defined definition denotes df(x directional derivative dpf(x dual pair equation Exercise exists a neighborhood extremal principle F-differentiable finite-dimensional Frechet smooth Banach function g Further let G-derivative graphs Hilbert space implies inequality intdom Lemma Let f liminf limsup locally convex locally convex spaces locally L-continuous lower semicontinuous mapping mean value theorem minimizer monotone Mordukhovich multifunction multiplier rule nonempty subset normal cone normed vector space numbers obtain optimality conditions problem Proof proper and convex Proposition Recall reflexive Banach space Remark result Rockafellar satisfying Sect smooth Banach space solution strong minimizer subdifferential mapping sufficiently small topological vector space topology uniformly convex verify zero