Convergence Analysis of Processes with Valiant Projection Operators in Hilbert Space

TL;DR

This paper introduces a new class of projection operators called valiant projectors for convex feasibility problems in Hilbert spaces, providing convergence analysis and extending previous relaxation methods.

Contribution

It proposes valiant projection operators that implement a continuous relaxation strategy, and proves their convergence in solving convex feasibility problems.

Findings

Valiant projection operators extend existing relaxation methods.

The convergence of the proposed method is rigorously proven.

The approach generalizes the 1985 automatic relaxation method of Censor.

Abstract

Convex feasibility problems require to find a point in the intersection of a finite family of convex sets. We propose to solve such problems by performing set-enlargements and applying a new kind of projection operators called valiant projectors. A valiant projector onto a convex set implements a special relaxation strategy, proposed by Goffin in 1971, that dictates the move toward the projection according to the distance from the set. Contrary to past realizations of this strategy, our valiant projection operator implements the strategy in a continuous fashion. We study properties of valiant projectors and prove convergence of our new valiant projections method. These results include as a special case and extend the 1985 automatic relaxation method of Censor.