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Taylor’s formula for C k,1 functions. (English) Zbl 0852.49012
The aim of the paper is to extend Taylor’s formula to C k,1 functions, i.e., functions whose kth 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.
49J52Nonsmooth analysis (other weak concepts of optimality)
26B25Convexity and generalizations (several real variables)
26B10Implicit function theorems, Jacobians, transformations with several real variables