SHARP VARIATIONAL CONDITIONS FOR CONVEX COMPOSITE NONSMOOTH FUNCTIONS.

In this paper, we present, first- and second-order variational conditions for a convex composite function go F, where g is a nonsmooth convex function and F is a vector valued map. The first-order results, which apply to (not, necessarily locally Lipsehitz) continuous maps F, not only recapture the results of the special cases where F is locally Lipschitz or Gâteaux differentiable but also yield sharp necessary variational conditions in these eases. The result, s arc achieved by applying a new strengthened notion of approximate Jacobian, called a Gâteaux (G) approximate Jacobian, without the use of the upper semicontinuity of the approximate Jacobian. These variational results are generally derived by using a chain rule formula or by constructing upper convex approximations to the composite function. These approaches often need the upper semicontinuity requirement of a generalized Jacobian map. Such a requirement not only limits the derivation of sharp optimality conditions, as the "small" approximate Jacobians (or generalized subdifferentials) lack an upper semi-continuity property, but also restricts the treatment of Gâteaux differentiable maps F. This situation is overcome by the use of G-approximate Jacobians. The second order variational conditions are shown to hold, in particular, in the case where F is continuously Gâteaux differentiable. [ABSTRACT FROM AUTHOR]