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On Sufficient Conditions in Nonsmooth Optimization

Second-order conditions are given which are sufficient for a point to be a local minimizer for certain nonlinear programming problems defined on Banach spaces. The functions involved in the problems are not required to be smooth or convex; indeed, they are required only to satisfy certain conditions which are weak enough to be satisfied by all locally Lipschitzian functions. The sufficiency conditions are expressed in terms of Clarke generalized gradients as extended by Rockafellar. An account is given of the connections between these results and some earlier sufficiency theorems.