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Convergence Analysis of Processes with Valiant Projection Operators in Hilbert Space

Yair Censor(yair***at***math.haifa.ac.il)
Rafiq Mansour(intogral***at***gmail.com)

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.

Keywords: Intrepid projector; convex feasibility problem; valiant projector; set-enlargement; automatic relaxation method (ARM); ART3 algorithm; Goffin's principle

Category 1: Convex and Nonsmooth Optimization (Convex Optimization )

Category 2: Nonlinear Optimization (Constrained Nonlinear Optimization )

Citation: December 4, 2016. Revised: May 29, 2017. Revised: September 11, 2017. Journal of Optimization Theory and Applications, accepted for publication.

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Entry Submitted: 09/15/2017
Entry Accepted: 09/16/2017
Entry Last Modified: 09/15/2017

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