Taylor’s formula for \(C^{k,1}\) functions. (English) Zbl 0852.49012

Dinh The Luc

SIAM J. Optim. 5, No. 3, 659-669 (1995).

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.

Reviewer: M.Studniarski (Łódź)

MSC:

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

Keywords:

Taylor’s formula; subdifferential; chain rule; quasiconvex functions