Abstract

We present first- and second-order variational conditions for a convex composite function \(g\circ F\), where \(g\) is a nonsmooth convex function and \(F\) is a vector-valued map. The first-order results, which apply to (not necessarily locally Lipschitz) 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 cases. The results are 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.