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.
Bibliographical noteFunding Information:
Acknowledgements We thank Tommy Elfving for reading several parts of this paper and making enlightening comments. We are indebted to the reviewers and to the Editor-in-Chief Franco Giannessi for their insightful and constructive comments that helped us improve the paper. This work was supported by Research Grant No. 2013003 of the United States-Israel Binational Science Foundation (BSF).
© 2017, Springer Science+Business Media, LLC.
- ART3 algorithm
- Automatic relaxation method (ARM)
- Convex feasibility problem
- Goffin’s principle
- Intrepid projector
- Valiant projector
ASJC Scopus subject areas
- Management Science and Operations Research
- Control and Optimization
- Applied Mathematics