Sharp Variational Conditions for Convex Composite Nonsmooth Functions
V. Jeyakumar, Dinh The Luc · SIAM Journal on Optimization · 2002
In this paper, 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.