Approximate Jacobian Matrices for Nonsmooth Continuous Maps and C¹-Optimization

Abstract

|A notion of approximate Jacobian matrix is introduced for a continuous vector-valued map. It is shown for instance that the Clarke generalized Jacobian is an approximate Jacobian for a locally Lipschitz map. The approach is based on the idea of convexi cators of real-valued functions. Mean value conditions for continuous vector-valued maps and Taylor's expansions for continuously Gâteaux di erentiable functions (i.e. C 1-functions) are presented in terms of approximate Jacobians and approximate Hessians respectively. Second-order necessary, and su cient conditions for optimality and convexity of C 1-functions are also given.

Inverse and implicit function theorems forH-differentiable and semismooth functions

“…Following Jeyakumar and Luc [19], we define the concept of an approximate Jacobian: given f : ⊆ R n → R m and x * ∈ , we say that a closed set ∂ * f (x * ) ⊆ R m×n is an approximate Jacobian of f at x * if for all v ∈ R m , the following inequality holds:…”

“…[16] that the Fréchet derivative of a Fréchet differentiable function, the Clarke generalized Jacobian of a locally Lipschitzian function [5], the Bouligand differential of a semismooth function [39], and the C-differential of a C-differentiable function [40] are examples of H -differentials. H -differentials are related to the approximate Jacobians of Jeyakumar and Luc [19], in that the closure of an H -differential is an approximate Jacobian. Applications of Hdifferentiability to optimization, complementarity, and variational inequalities are treated in Ref.…”

Generalized Hessians of $C^{1,1}$-Functions and Second-Order Viscosity Subjets

“…In the literature of nonsmooth analysis there are many works on this topic, see for example [8], [11], [16] and references therein. See also [15], [19], [20] for applications to positive semidefinite optimization, as well as [12], [13] for some generalizations. In this setting, a second-order result -analogous to (3)-reads as follows:…”

Necessary optimality conditions for bilevel set optimization problems

“…Very recently, as an extension of the notion of subdifferentials, the idea of convexificators has been used to extend, unify, and sharpen various results in nonsmooth analysis and optimization [11,23,24]. In [25], Jeyakumar and Luc gave a revised version of convexificators by introducing the notion of a convexificator which is a closed set but is not necessarily bounded or convex.…”