Approximate Jacobian Matrices for Nonsmooth Continuous Maps and C1-Optimization

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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 Gâteaux differentiable functions (i.e., C1-functions) are presented in terms of approximate Jacobians and approximate Hessians, respectively. Second-order necessary and sufficient conditions for optimality and convexity of C1-functions are also given.