Convergence Analysis of Processes with Valiant Projection Operators in Hilbert Space

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.

Asynchronous sequential inertial iterations for common fixed points problems with an application to linear systems

“…The ASI framework can be applied in conjunction with other projection methods. These include the valiant projection method (VPM) of Censor and Mansour [20], which is known as the automatic relaxation method (ARM) of Censor [10] when applied to interval linear inequalities. We conjecture the intrepid method of Bauschke, Iorio, and Koch [7], which is known as the ART3 method of Herman [37] when applied to linear systems, may also be used within the ASI framework.…”

Algorithms and Convergence Results of Projection Methods for Inconsistent Feasibility Problems: A Review

“…They defined an operator which they called the "intrepid projector", intended to generalize the ART3 algorithm of [50] to convex sets. Motivated by [14], Censor and Mansour [31] present a new operator, called the "valiant operator", that enables to implement the algorithmic principle embodied in the ARM of [20] to general convex feasibility problems. Both ART3 and ARM seek a feasible point in the intersections of the hyperslabs and so their generalizations to the convex case seek feasibility of appropriate enlargement sets that define the extended problem.…”