In this paper we establish conditions that guarantee, in the setting of a general Banach space, the Painlevé-Kuratowski convergence of the graphs of the subdifferentials of convexly composite functions. We also provide applications to the convergence of multipliers of families of constrained optimization problems and to the generalized second-order derivability of convexly composite functions.

When an optimization problem is represented by its essential objective function, which incorporates constraints through infinite penalties, first- and secondorder conditions for optimality can be stated in terms of the first- and second-order epi-derivatives of that function. Such derivatives also are the key to the formulation of subproblems determining the response of a problem's solution when the data values on which the problem depends are perturbed. It is vital for such reasons to have available a calculus of epi-derivatives. This paper builds on a central case already understood, where the essential objective function is the composite of a convex function and a smooth mapping with certain qualifications, in order to develop differentiation rules covering operations such as addition of functions and a more general form of composition. Classes of "amenable" functions are introduced to mark out territory in which this sharper form of nonsmooth analysis can be carried out.

In this paper we tackle the problem of identifying set-valued mappings that are subgradient set-valued mappings. We show that a set-valued mapping is the proximal subgradient mapping of a lower semicontinuous function bounded below by a quadratic if and only if it satisfies a monotone selection property.

Set-valued mappings arise quite naturally in optimization and nonsmooth analysis. In optimization, typically one has a family of optimization problems that depend on some parameter. One can then associate to this family of problems the set-valued mappings that assign to the parameter the set of optimal solutions, the set of feasible solutions or the set of multipliers. Many of these set-valued mappings encountered in optimization have been shown to be “proto-differentiable” (see Rockafellar [16]) i.e., in some sense these set-valued mappings are “differentiable”. Using estimates provided by the proto-derivatives, see Proposition 2.1, one can then obtain information on how the sets depend on the parameter. The concept of proto-differentiation is described in Section 2.

The Rank Theorem is proved for locally Lipschitz continuous functions f:Rn → Rp with generalized derivative of constant rank.