Prox-Regularity and Stability of the Proximal Mapping

Fundamental insights into the properties of a function come from the study of its Moreau envelopes and Proximal point mappings. In this paper we examine the stability of these two objects under several types of perturbations. In the simplest case, we consider tilt-perturbations, i.e. perturbations which correspond to adding a linear term to the objective function. We show that for functions that have single-valued Lipschitz continuous proximal mappings, in particular for prox-regular functions, tilt-perturbations result in stable, i.e. single-valued Lipschitz continuous, proximal point mappings. In the more complex case, we consider the class of parametrically prox-regular functions. These include most of the functions that arise in the framework of nonlinear programming and its extensions (e.g. convex, lower-

, strongly amenable (convexly composite)). New characterizations of prox-regularity are given and more general perturbations along the lines of A. B. Levy, R. A. Poliquin and R. T. Rockafellar [SIAM J. Optimization 10(2) (2000) 580–604] are studied. We show that under suitable conditions (compatible parameterization, positive coderivative,...), the proximal point mappings of the function

f_u(x) = f(x, u) f u ( x ) = f ( x , u )

depends in a Lipschitz fashion on the parameter u and the prox-parameter r.