Convergence of the associated sequence of normal cones of a Mosco convergent sequence of sets

摘要

In 1977, Attouch established a relationship between the Mosco epiconvergence of a sequence of convex functions and the graph convergence of the associated sequence of subdifferentials, which has been found to have many important applications in optimization. In 1995, Levy, Poliquin, and Thibault proved Attouch-type theorems for a sequence of primal-lower-nice (not necessarily convex) functions with variational behavior of "order two." In this paper, using the proximal analysis technique, we provide convergence results on the associated sequence of normal cones of a Moscoconvergent sequence of general closed sets. Under the uniform subsmoothness assumption, some sharper convergence results are established. As applications, we establish Attouch-type theorems for a sequence of more general quasi-subsmooth functions with variational behavior of "order one."