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.


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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