This book treats various concepts of generalized derivatives and subdifferentials in normed spaces, their geometric counterparts and their application to optimization problems. It starts with the subdifferential of convex analysis, passes to corresponding concepts for locally Lipschitz continuous functions and then presents subdifferentials for general lower semicontinuous functions. All basic tools are presented where they are needed: this concerns separation theorems, variational and extremal principles as well as relevant parts of multifunction theory. Each chapter ends with bibliographic notes and exercises.
49 pages matching inequality in this book
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The Conjugate of Convex Functionals
12 other sections not shown
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