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Strong convergence of projection-like methods in Hilbert spaces

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Abstract

Many mathematical and applied problems can be reduced to finding a common point of a system of convex sets. The aim of this paper is twofold: first, to present a unified framework for the study of all the projection-like methods, both parallel and serial (chaotic, mostremote set, cyclic order, barycentric, extrapolated, etc.); second, to establish strong convergence results for quite general sets of constraints (generalized Slater, generalized uniformly convex, made of affine varieties, complementary, etc.). This is done by introducing the concept of regular family. We proceed as follows: first, we present definitions, assumptions, theorems, and conclusions; thereafter, we prove them.

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References

  1. Gastinel, N.,Linear Numerical Analysis, Hermann, Paris, France, 1970.

    Google Scholar 

  2. Haugazeau, Y.,Sur les Inéquations Variationnelles et la Minimization des Fonctionnelles Convexes, University of Paris, Doctoral Thesis, 1968.

  3. Pierra, G.,Decomposition through Formalization in a Product Space, Mathematical Programming, Vol. 28, pp. 96–115, 1984.

    Google Scholar 

  4. Teman, R.,Quelques Méthodes de Décomposition en Analyse Numérique, Actes du Congrès International des Mathèmatiques, Gauthier Villars, Paris, France, pp. 311–319, 1971.

  5. Kaczmarz, S.,Angenäherte Auffösung von Systemen Linearer Gleichungen, Bulletin de l'Académie de Sciences de Pologne, Series A, Vol. 35, pp. 355–357, 1937.

    Google Scholar 

  6. Cimmino, G.,Galcolo Approximato per le Soluzioni dei Sistemi di Equazioni Lineari, La Ricerca Scientifica, Vol. 1, pp. 326–333, 1938.

    Google Scholar 

  7. Agmon, S.,The Relaxation Method for Linear Inequalities, Canadian Journal of Mathematics, Vol. 6, pp. 382–392, 1954.

    Google Scholar 

  8. Ermin, I. I.,Generalization of the Motskin-Agmon Relaxation Method, Uspekhi Matematicheskie Nauk, Vol. 20, pp. 183–199, 1965.

    Google Scholar 

  9. Motskin, T. S., andShoenberg, I.,The Relaxation Method for Linear Inequalities, Canadian Journal of Mathematics, Vol. 6, pp. 393–404, 1954.

    Google Scholar 

  10. Tanabe, K.,Projection Method for Solving a Singular System of Linear Equations and Its Applications, Numerische Mathematik, Vol. 17, pp. 203–214, 1971.

    Google Scholar 

  11. Censor, Y., andElfring, T.,New Methods for Linear Inequalities, University of Haifa, Department of Mathematics, Report No. 15, 1981.

  12. Vasilchenko, andSvetlakov, A. A.,Projection Algorithm for Solving System of Linear Equations of High Dimensionality, Computational Mathematics and Mathematical Physics, Vol. 20, pp. 1–8, 1980.

    Google Scholar 

  13. Levitin, E. S., andPolyak, B. T.,Method of Minimizing under Restrictions, Zhurnal Vychislitelnoi Matematiki i Matematicheskoi Fiziki, Vol. 6, pp. 787–789, 1966.

    Google Scholar 

  14. Merzlyakov, Y. I.,On the Relaxation Method for Solving Systems of Linear Inequalities, Zhurnal Vychislitelnoi Matematiki i Matematicheskoi Fiziki, Vol. 2, pp. 482–487, 1962.

    Google Scholar 

  15. Auslender, A.,Méthode Numérique pour la Résolution des Problèmes d'Optimisation avec Contraintes, University of Grenoble, Doctoral Thesis, 1969.

  16. Oettli, W.,Symmetric Duality and a Convergent Subgradient Method for Discrete Linearly Constrained Approximation Problems with Arbitrary Norms Appearing in the Objective Function and the Constraints, Journal of Approximation Theory, Vol. 14, pp. 43–50, 1975.

    Google Scholar 

  17. Pierra, G.,Méthode de Décomposition et de Croisement d'Algorithmes pour les Problèmes d'Optimization, University of Grenoble, Doctoral Thesis, 1976.

  18. Pierra, G.,Eclatement des Contraintes en Parallèle pour la Minimization d'une Forme Quadratique, Lecture Notes in Computer Science 41, Edited byJ. Cea, Springer-Verlag, Berlin, Germany, pp. 200–218, 1976.

    Google Scholar 

  19. Martinet, B.,Algorithmes pour la Résolution des Problèmes d'Optimisation et de Minimax, University of Grenoble, Doctoral Thesis, 1972.

  20. Kammerer, W. J., andNashmed, M. Z.,A Generalization of a Matrix Iterative Method of G. Cimmino to Best Approximation: Solution of Linear Integral Equations of the First Kind, Rendiconti Accademia Nazionale dei Lincei, Vol. 51, pp. 20–25, 1971.

    Google Scholar 

  21. Altschuller, D., andCensor, Y.,Mathematical Aspects of Image Reconstruction from Projections, Progress in Pattern Recognition, Edited by L. N. Kanal and A. Rosenfeld, North-Holland, Amsterdam, Holland, 1981.

    Google Scholar 

  22. Herman, G. T.,Image Reconstruction from Projections, Fundamentals of Computerized Tomography, Academic Press, New York, New York, 1980.

    Google Scholar 

  23. Herman, G. T., Lent, A., andLutz, P. H.,Relaxation Methods for Image Reconstruction, Communications of the Association for Computing Machinery, Vol. 21, pp. 152–158, 1978.

    Google Scholar 

  24. Gubin, L. G., Polyak, B. T., andRaik, E. V.,The Method of Projections for Finding the Common Point of Convex Sets, Computational Mathematics and Mathematical Physics, Vol. 6, pp. 1–24, 1967.

    Google Scholar 

  25. Bergman, L. H.,Finding the Common Point of Convex Sets by the Method of Successive Projection, Doklady Akademy Nauk SSSR, Vol. 16, pp. 487–490, 1965.

    Google Scholar 

  26. Robert, F.,Contraction en Norme Vectorielle, Convergence d'Itérations Chaotiques pour des Equations non Linéaires de Point Fixe à Plusieurs Variables, Linear Algebra and Its Applications, Vol. 13, pp. 19–35, 1976.

    Google Scholar 

  27. Charnay, M.,Itérations Chaotiques sur un Produit d'Espèces Métriques. Université Claude Bernard, Lyon, Doctoral Thesis, 1975.

  28. Chazan, D., andMiranker, W.,Chaotic Relaxation, Linear Algebra and Its Applications, Vol. 2, pp. 199–222, 1969.

    Google Scholar 

  29. Baudet, G. M.,Asynchrons Iterative Methods for Multiprocessors, Journal of the Association for Computing Machinery, Vol. 25, pp. 226–244, 1978.

    Google Scholar 

  30. Kung, H. T.,Synchronized and Asynchronous Parallel Algorithms for Multiprocessors, Algorithms and Complexity: New Directions and Recent Results, Edited by J. F. Trub, Academic Press, New York, New York, 1976.

    Google Scholar 

  31. Censor, Y., andLent, A.,Cyclic Subgradient Projections, University of Haifa, Department of Mathematics, Report No. 35, 1981.

  32. Bourbaki, N.,Espace Vectoriel Topologique, Hermann, Paris, France, 1973.

    Google Scholar 

  33. Deny, J.,Cours de Mathématiques Topologie, Hermann, Paris, France, 1971.

    Google Scholar 

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Authors and Affiliations

  1. Ecole Nationale Supérieure de Mécanique et d'Aérotechnique, University of Poitiers, Poitiers, France

    N. Ottavy (Assistant Professor)

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  1. N. Ottavy
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Communicated by E. Polak

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Ottavy, N. Strong convergence of projection-like methods in Hilbert spaces. J Optim Theory Appl 56, 433–461 (1988). https://doi.org/10.1007/BF00939552

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