Generalized projections onto convex sets

Abstract

This paper introduces the notion of projection onto a closed convex set associated with a convex function. Several properties of the usual projection are extended to this setting. In particular, a generalization of Moreau's decomposition theorem about projecting onto closed convex cones is given. Several examples of distances and the corresponding generalized projections associated to particular convex functions are presented.

The Representation and Continuity of a Generalized Metric Projection onto a Closed Hyperplane in Banach Spaces

“…In fact, let ∈ −1 ( * ). Then ( ) = ∘ ( * ) by (13). This, together with Proposition 1(vi), implies that ‖ ‖ ≤ (1/ 1 ) ( ) = ( ∘ ( * )/ 1 ), and (44) is proved.…”

“…This together with (13) implies that ∘ ( * ) 0 ∈ −1 ( * ), and therefore −1 ( * ) ̸ = 0. By Theorem 6, one sees that ( * , ) ( ) ̸ = 0.…”

“…Hence, 1 + (1 − ) 2 ∈ −1 ( * ) by (13), and −1 ( * ) is convex. We then show that −1 ( * ) is weakly compact.…”

“…When is nearly strictly convex, Wang has shown in [11] that ( ) ( * , ) is norm-to-weak upper semicontinuous on , while, when is arbitrary Banach space, Zhang and Shi have given in [12] the pointwise continuity of ( ) ( * , ) under an additional condition. It should be noted that, when one uses a nonnegative convex function on the Euclidean space R satisfying (0) = 0 and ( ) = ( ) for all ∈ R and ≥ 0 as a metric on R (i.e., the distance from a point to a subset of R is defined as ( , ) = inf ∈ ( − )), Ferreia and Nemeth have investigated in [13] the problem of the best approximation in R and, in particular, given some properties of corresponding metric projections on a hyperplane in R .…”