Aspects of the Projector on Prox-Regular Sets

Abstract

This paper deals with results about projectors on the important class of closed, prox-regular sets in ℝⁿ. These sets, which include all closed convex sets but also many nonconvex sets, have the property that their associated projection mappings are very well behaved, being locally single-valued and continuous among other good properties. We give elementary proofs of these properties of the projector, and for the case in which the projection is made onto a perturbed set we show that under suitable conditions the projector is jointly continuous in the perturbation variable and the variable expressing the point that is projected. We briefly describe an application to the extension of a normal-map construction from variational inequalities posed over polyhedral convex sets to variational conditions posed over sets that satisfy prox-regularity.

The research reported here was sponsored in part by the National Science Foundation under Grant DMS-0305930, in part by the U. S. Army Research Office under Grant DAAG19-01-1-0502, and in part by the Air Force Research Laboratory under agreement number F49620-01-1-0040. The U. S. Government has certain rights in this material, and is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the author and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the sponsoring agencies or the U. S. Government.