Continuity Properties of Projection Operators

Many optimization problems can be reformulated as best approximation problems with respect to an appropriate norm. Their solutions are thus given by projection operators. It follows that projection operators play a key role in several areas, such as mechanics, minimization algorithms, variational inequalities, complementary problems. Thus, it is of interest to study the properties of such operators. In particular, continuity properties of projection maps express important properties about the dependence on parameters of the solution maps of such problems. It is well known that the projection operator onto a nonempty closed convex subset of a uniformly convex Banach space is continuous (see, e.g., [5, 19] and the references in the discussion closing the paper). We prove here that it is uniformly continuous on bounded sets and we provide a simple estimate of the modulus of uniform continuity. We also show that the projection of a given point onto a nonempty closed convex subset C is continuous with respect to C for the topology defined by the Hausdorff-Pompeiu metric, or, more precisely, the family of bounded hemi-metrics of Hausdorff-Pompeiu type used in [7, 8, 9, 10, 28, 29, 35] which defines a more realistic uniform structure. The associated convergence is metrizable and it has been the subject of many recent studies; see [36] for a recent survey and references. It is known that even in Hilbert spaces this dependence is of Hölder type and cannot be of Lipschitz type [8, 13]. Note however, that any uniformly continuous mapping satisfies a Lipschitz type inequality for points which are sufficiently apart (see [32] for a recent proof). Here we deal with the case of uniformly convex Banach spaces since this framework is adapted to questions of existence and uniqueness. We use a general duality mapping,