Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A norm convergence result on random products of relaxed projections in Hilbert space

Author: H. H. Bauschke
Journal: Trans. Amer. Math. Soc. 347 (1995), 1365-1373
MSC: Primary 47H09; Secondary 46C99, 47N99, 92C55, 94A12
MathSciNet review: 1257097
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Suppose $X$ is a Hilbert space and ${C_1}, \ldots ,{C_N}$ are closed convex intersecting subsets with projections ${P_1}, \ldots ,{P_N}$. Suppose further $r$ is a mapping from $\mathbb {N}$ onto $\{ 1, \ldots ,N\}$ that assumes every value infinitely often. We prove (a more general version of) the following result: If the $N$-tuple $({C_1}, \ldots ,{C_N})$ is "innately boundedly regular", then the sequence $({x_n})$, defined by \[ {x_0} \in X\;{\text {arbitrary,}}\quad {x_{n + 1}}: = {P_{r(n)}}{x_n},\quad {\text {for all}}\;n \geqslant 0,\] converges in norm to some point in $\cap _{i = 1}^N{C_i}$. Examples without the usual assumptions on compactness are given. Methods of this type have been used in areas like computerized tomography and signal processing.

References [Enhancements On Off] (What's this?)

  • Shmuel Agmon, The relaxation method for linear inequalities, Canad. J. Math. 6 (1954), 382–392. MR 62786, DOI
  • Ron Aharoni and Yair Censor, Block-iterative projection methods for parallel computation of solutions to convex feasibility problems, Proceedings of the Fourth Haifa Matrix Theory Conference (Haifa, 1988), 1989, pp. 165–175. MR 1010054, DOI
  • I. Amemiya and T. Andô, Convergence of random products of contractions in Hilbert space, Acta Sci. Math. (Szeged) 26 (1965), 239–244. MR 187116
  • Jean-Bernard Baillon and Ronald E. Bruck, Ergodic theorems and the asymptotic behavior of contraction semigroups, Fixed point theory and applications (Halifax, NS, 1991) World Sci. Publ., River Edge, NJ, 1992, pp. 12–26. MR 1190029
  • J. B. Baillon, R. E. Bruck, and S. Reich, On the asymptotic behavior of nonexpansive mappings and semigroups in Banach spaces, Houston J. Math. 4 (1978), no. 1, 1–9. MR 473932
  • H. H. Bauschke and J. M. Borwein, On projection algorithms for solving convex feasibility problems, Technical rep., Simon Fraser Univ., 1993.
  • Ronald E. Bruck, Random products of contractions in metric and Banach spaces, J. Math. Anal. Appl. 88 (1982), no. 2, 319–332. MR 667060, DOI
  • Yair Censor, Parallel application of block-iterative methods in medical imaging and radiation therapy, Math. Programming 42 (1988), no. 2, (Ser. B), 307–325. MR 976123, DOI
  • Y. Censor and G. T. Herman, On some optimization techniques in image reconstruction from projections, Appl. Numer. Math. 3 (1987), 365-391. F. Deutsch, The method of alternating orthogonal projections, Approximation Theory, Spline Features and Applications (S. P. Singh, ed.), Proc. Conf., Hotel Villa del Mare, Maratea, Italy, April 28, 1991, May 9, 1991, Kluwer Academic, Amsterdam, 1992, pp. 105-121. J. M. Dye, Convergence of random products of compact contractions in Hilbert space, Integral Equations and Operator Theory 12 (1989), 12-22. J. M. Dye, T. Kuczumow, P.-K. Lin, and S. Reich, Random products of nonexpansive mappings in spaces with the Opial property, (B.-L. Lin and W. B. Johnson, eds.), Banach Spaces, Contemporary Math., vol. 144, Amer. Math. Soc., Providence, RI, 1993, pp. 87-93. J. M. Dye and S. Reich, Random products of nonexpansive mappings, Optimization and Nonlinear Analysis (A. Ioffe, M. Marcus, and S. Reich, eds.), Proc. Binational Workshop on Optimization and Nonlinear Analysis, Technion City, Haifa, 21-27, March 1990, Pitman Res. Notes in Math. Ser., vol. 244, Longman Sci. Tech, Harlow, England, 1992, pp. 106-118. ---, Unrestricted iterations of nonexpansive mappings in Hilbert space, Nonlinear Anal. 18 (1992), 199-207. L. Elsner, I. Koltracht, and M. Neumann, Convergence of sequential and asynchronous nonlinear paracontractions, Numer. Math. 62 (1992), 305-319. S. D. Flåm and J. Zowe, Relaxed outer projections, weighted averages and convex feasibility, BIT 30 (1990), 289-300. A. Genel and J. Lindenstrauss, An example concerning fixed points, Israel J. Math. 22 (1975), 81-86. K. Goebel and W. A. Kirk, Topics in metric fixed point theory, Cambridge Stud. Adv. Math., vol. 28, Cambridge Univ. Press, Cambridge, 1990. K. Goebel and S. Reich, Uniform convexity, hyperbolic geometry and nonexpansive mappings, Monographs and Textbooks in Pure and Appl. Math., vol. 83, Marcel Dekker, New York, 1984. L. G. Gubin, B. T. Polyak, and E. V. Raik, The method of projections for finding the common point of convex sets, USSR Comput. Math. Math. Phys. 7 (1967), 1-24. I. Halperin, The product of projection operators, Acta Sci. Math. (Szeged) 23 (1962), 96-99. S. Kaczmarz, Angenäherte Auflösung von Systemen linearer Gleichungen, Bull. Internat. Acad Polon. Sci. Lettres. Cl. Sci. Math. Natur. Sér. A: Sci. Math., Imprimerie de l’Université, Cracovie, 1937, pp. 355-357. T. S. Motzkin and I. J. Schoenberg, The relaxation method for linear inequalities, Canad. J. Math. 6 (1954), 393-404. W. V. Petryshyn and T. E. Williamson, Jr., A necessary and sufficient condition for the convergence of a sequence of iterates for quasi-nonexpansive mappings, Bull. Amer. Math. Soc. 78 (1972), 1027-1031. ---, Strong and weak convergence of the sequence of successive approximations for quasi-nonexpansive mappings, J. Math. Anal. Appl. 43 (1973), 459-497. M. I. Sezan, An overview of convex projections theory and its applications to image recovery problems, Ultramicroscopy 40 (1992), 55-67. P. Tseng, On the convergence of the products of firmly nonexpansive mappings, SIAM J. Optim. 2 (1992), 425-434. J. von Neumann, Functional operators, vol. II. The geometry of orthogonal spaces, Ann. of Math. Stud., no. 22, Princeton Univ. Press, Princeton, NJ, 1950. Reprint of mimeographed lecture notes first distributed in 1933. D. C. Youla, On deterministic convergence of iterations of relaxed projection operators, J. Visual Communication and Image Representation 1 (1990), 12-20. D. C. Youla and H. Webb, Image reconstruction by the method of convex projections: Part $1$. Theory, IEEE Trans. Medical Imaging MI-1 (1982), 81-94.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 47H09, 46C99, 47N99, 92C55, 94A12

Retrieve articles in all journals with MSC: 47H09, 46C99, 47N99, 92C55, 94A12

Additional Information

Keywords: Banach contraction, computerized tomography, convex feasibility problem, convex programming, convex set, Fejér monotone sequence, Hilbert space, image reconstruction, image recovery, innate bounded regularity, Kaczmarz’s method, nonexpansive mapping, orthogonal projection, projection algorithm, projection method, projective mapping, random product, relaxation method, relaxed projection, signal processing, unrestricted iteration, unrestricted product
Article copyright: © Copyright 1995 American Mathematical Society