We consider minimization of functions that are compositions of convex or prox-regular functions (possibly extended-valued) with smooth vector functions. A wide variety of important optimization problems fall into this framework. We describe an algorithmic framework based on a subproblem constructed from a linearized approximation to the objective and a regularization term. Properties of local solutions of this subproblem underlie both a global convergence result and an identification property of the active manifold containing the solution of the original problem. Preliminary computational results on both convex and nonconvex examples are promising.
We introduce a new class of multifunctions whose graphs under certain "kernel inverting" matrices, are locally equal to the graphs of Lipschitzian (single-valued) mappings. We characterize the existence of Lipschitzian localizations of these multifunctions in terms of a natural condition on a generalized Jacobian mapping. One corollary to our main result is a Lipschitzian inverse mapping theorem for the broad class of "max hypomonotone" multifunctions. We apply our theoretical results to the sensitivity analysis of solution mappings associated with parameterized optimization problems. In particular, we obtain new characterizations of the Lipschitzian stability of stationary points and Karush-Kuhn-Tucker pairs associated with parameterized nonlinear programs.