Approximate Jacobian matrices for nonsmooth continuous maps and C¹-optimization

Abstract

The notion of approximate Jacobian matrices 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 convexificators of real valued functions. Mean value conditions for continuous vector-valued maps and Taylor’s expansions for continuously Gˆateaux differentiable functions (i.e., C1-functions) are presented in terms of ap proximate Jacobians and approximate Hessians, respectively. Second-order necessary and sufficient conditions for optimality and convexity of C1-functions are also given