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Zbl 0852.49012
Dinh The Luc
(Dinh The Luc)
Taylor's formula for $C\sp{k,1}$ functions. (English)
[J] SIAM J. Optim. 5, No.3, 659-669 (1995). ISSN 1052-6234; ISSN 1095-7189/e

The aim of the paper is to extend Taylor's formula to $C^{k, 1}$ functions, i.e., functions whose $k$th order derivatives are locally Lipschitz. First, the author defines the $(k+ 1)$th order subdifferential of a $C^{k, 1}$ function and gives a chain rule for this subdifferential. Then, two versions of Taylor's theorem are established. A calculus rule for generalized Hessian of implicit functions is also presented. The results are then applied to derive high-order optimality conditions and second-order characterizations of quasiconvex functions.
[M.Studniarski (Łódź)]

MSC 2000:: *49J52 Nonsmooth analysis (other weak concepts of optimality)
26B25 Convexity and generalizations (several real variables)
26B10 Implicit function theorems, etc. (several real variables)

Keywords: Taylor's formula; subdifferential; chain rule; quasiconvex functions

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