Abstract. In this paper, we present a generalization of the Hessian matrix to C L1 functions, i.e., to functions whose gradient mapping is locally Lipschitz. This type of function arises quite naturally in nonlinear analysis and optimization. First the properties of the generalized Hessian matrix are investigated and then some calculus rules are given. In particular, a second-order Taylor expansion of a C 1'1 function is derived. This allows us to get second-order optimality conditions for nonlinearly constrained mathematical programming problems with C ~'~ data.

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