Nonsmooth AnalysisThis 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. |
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analogous Asplund space assertion Assume assumptions chain rule closed subset continuously differentiable convergent convex cone convex functional convex set convex subset Corollary defined definition denotes directional derivative dual pair epi ƒ equation equivalent Exercise exists a neighborhood extremal principle f is continuous F-differentiable finite-dimensional Fréchet smooth Banach functional ƒ Further let G-derivative graph Hence Hilbert space implies inequality int dom f Lemma Let f lim inf lim sup Lipschitz locally convex locally convex spaces locally L-continuous lower semicontinuous mapping mean value theorem minimizer of ƒ monotone Mordukhovich multifunction nonempty subset normal cone normed vector space numbers obtain optimality conditions problem Proof proper and convex Proposition Recall Remark satisfying sequence smooth Banach space solution strong minimizer subdifferential subdifferential mapping sufficiently small Tc(A topology verify zero