## 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 $$k$$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.

### MSC:

 49J52 Nonsmooth analysis 26B25 Convexity of real functions of several variables, generalizations 26B10 Implicit function theorems, Jacobians, transformations with several variables
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