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Iterative methods for approximating fixed points of Bregman nonexpansive operators
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August  2013, 6(4): 1043-1063. doi: 10.3934/dcdss.2013.6.1043

## Iterative methods for approximating fixed points of Bregman nonexpansive operators

 1 Departamento de Análisis Matemático 2 Universidad de Sevilla 3 Apdo. 1160, 41080 Sevilla 4 Department of Mathematics 5 The Technion-Israel Institute of Technology 6 32000 Haifa 7 The Technion --- Israel Institute of Technology

Received  July 2011 Revised  March 2012 Published  December 2012

Diverse notions of nonexpansive type operators have been extended to the more general framework of Bregman distances in reflexive Banach spaces. We study these classes of operators, mainly with respect to the existence and approximation of their (asymptotic) fixed points. In particular, the asymptotic behavior of Picard and Mann type iterations is discussed for quasi-Bregman nonexpansive operators. We also present parallel algorithms for approximating common fixed points of a finite family of Bregman strongly nonexpansive operators by means of a block operator which preserves the Bregman strong nonexpansivity. All the results hold, in particular, for the smaller class of Bregman firmly nonexpansive operators, a class which contains the generalized resolvents of monotone mappings with respect to the Bregman distance.
Citation: Victoria Martín-Márquez, Simeon Reich, Shoham Sabach. Iterative methods for approximating fixed points of Bregman nonexpansive operators. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1043-1063. doi: 10.3934/dcdss.2013.6.1043
##### References:
 [1] R. Aharoni and Y. Censor, Block-iterative projection methods for parallel computation of solutions to convex feasibility problems,, Linear Algebra Appl., 120 (1989), 165. doi: 10.1016/0024-3795(89)90375-3. Google Scholar [2] H. H. Bauschke, J. M. Borwein and P. L. Combettes, Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces,, Comm. Contemp. Math., 3 (2001), 615. doi: 10.1142/S0219199701000524. Google Scholar [3] H. H. Bauschke, J. M. Borwein and P. L. Combettes, Bregman monotone optimization algorithms,, SIAM J. Control Optim., 42 (2003), 596. doi: 10.1137/S0363012902407120. Google Scholar [4] H. H. Bauschke and P. L. Combettes, "Convex Analysis and Monotone Operator Theory in Hilbert Spaces,", Springer, (2011). doi: 10.1007/978-1-4419-9467-7. Google Scholar [5] H. H. Bauschke, E. Matoušková and S. Reich, Projection and proximal point methods: convergence results and counterexamples,, Nonlinear Anal., 56 (2004), 715. doi: 10.1016/j.na.2003.10.010. Google Scholar [6] V. Berinde, "Iterative Approximation of Fixed Points,", $2^{nd}$, (1912). Google Scholar [7] J. M. Borwein, S. Reich and S. Sabach, A characterization of Bregman firmly nonexpansive operators using a new monotonicity concept,, J. Nonlinear Convex Anal., 12 (2011), 161. Google Scholar [8] J. F. Bonnans and A. Shapiro, "Perturbation Analysis of Optimization Problems,", Springer, (2000). Google Scholar [9] L. M. Bregman, The method of successive projection for finding a common point of convex sets,, Soviet Math. Dokl., 6 (1965), 688. Google Scholar [10] L. M. Bregman, The relaxation method for finding a common point of convex sets and its application to the solution of problems in convex programming,, USSR Comput. Math. Math. Phys., 7 (1967), 200. Google Scholar [11] R. E. Bruck and S. Reich, Nonexpansive projections and resolvents of accretive operators in Banach spaces,, Houston J. Math., 3 (1977), 459. Google Scholar [12] D. Butnariu and Y. Censor, Strong convergence of almost simultaneous block-iterative projection methods in Hilbert spaces,, J. Comput. Appl. Math., 53 (1994), 33. doi: 10.1016/0377-0427(92)00123-Q. Google Scholar [13] D. Butnariu, Y. Censor and S. Reich, Iterative averaging of entropic projections for solving stochastic convex feasibility problems,, Comput. Optim. Appl., 8 (1997), 21. doi: 10.1023/A:1008654413997. Google Scholar [14] D. Butnariu and A. N. Iusem, "Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization,", Kluwer Academic Publishers, (2000). doi: 10.1007/978-94-011-4066-9. Google Scholar [15] D. Butnariu and E. Resmerita, Bregman distances, totally convex functions and a method for solving operator equations in Banach spaces,, Abstr. Appl. Anal., 2006 (2006), 1. doi: 10.1155/AAA/2006/84919. Google Scholar [16] Y. Censor, Row-action methods for huge and sparse systems and their applications,, SIAM Rev., 23 (1981), 444. doi: 10.1137/1023097. Google Scholar [17] Y. Censor and A. Lent, An iterative row-action method for interval convex programming,, J. Optim. Theory Appl., 34 (1981), 321. doi: 10.1007/BF00934676. Google Scholar [18] Y. Censor and S. Reich, Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization,, Optimization, 37 (1996), 323. doi: 10.1080/02331939608844225. Google Scholar [19] C. Chidume, "Geometric Properties of Banach Spaces and Nonlinear Iterations,", Lecture Notes in Mathematics 1965, (1965). Google Scholar [20] G. Cimmino, Calcolo approssimato per le soluzioni dei sistemi di equazioni lineari,, Ric. Sci. (Roma), 9 (1938), 326. Google Scholar [21] J. Dugundji and A. Granas, "Fixed Point Theory,", Springer, (2003). Google Scholar [22] A. Genel and J. Lindenstrauss, An example concerning fixed points,, Israel J. Math., 22 (1975), 81. Google Scholar [23] K. Goebel and S. Reich, "Uniform Convexity, Hyperbolic Ggeometry, and Nonexpansive Mappings,", Marcel Dekker, (1984). Google Scholar [24] S. Kaczmarz, Angenäherte Auflösung von Systemen linearer Gleichungen,, Bull. Internat. Acad. Polon. Sci. Lett. Sér. A Sci. Math., 35 (1937), 355. Google Scholar [25] G. Kassay, S. Reich and S. Sabach, Iterative methods for solving systems of variational inequalities in reflexive Banach spaces,, SIAM J. Optim., 21 (2011), 1319. doi: 10.1137/110820002. Google Scholar [26] M. Kikkawa and W. Takahashi, Approximating fixed points of nonexpansive mappings by the block iterative method in Banach spaces,, Int. J. Comput. Numer. Anal. Appl., 5 (2004), 59. Google Scholar [27] M. A. Krasnosel'ski, Two observations about the method of successive approximations,, Uspehi Math. Nauk., 10 (1955), 123. Google Scholar [28] M. A. Krasnosel'ski and P. P. Zabreiko, "Geometrical Methods of Nonlinear Analysis,", Springer, (1984). doi: 10.1007/978-3-642-69409-7. Google Scholar [29] W. R. Mann, Mean value methods in iteration,, Proc. Amer. Math. Soc., 4 (1953), 506. Google Scholar [30] S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces,, J. Math. Anal. Appl., 67 (1979), 274. doi: 10.1016/0022-247X(79)90024-6. Google Scholar [31] S. Reich, A weak convergence theorem for the alternating method with Bregman distances,, in, (1996), 313. Google Scholar [32] S. Reich and S. Sabach, A strong convergence theorem for a proximal-type algorithm in reflexive Banach spaces,, J. Nonlinear Convex Anal., 10 (2009), 471. Google Scholar [33] S. Reich and S. Sabach, Two strong convergence theorems for Bregman strongly nonexpansive operators in reflexive Banach spaces,, Nonlinear Analysis, 73 (2010), 122. doi: 10.1016/j.na.2010.03.005. Google Scholar [34] S. Reich and S. Sabach, Existence and approximation of fixed points of Bregman firmly nonexpansive mappings in reflexive Banach spaces,, in, (2011), 299. doi: 10.1007/978-1-4419-9569-8_15. Google Scholar [35] E. Zeidler, "Nonlinear Functional Analysis and Its Applications, Vol. I, Fixed-Point Theorems,", Springer, (1986). doi: 10.1007/978-1-4612-4838-5. Google Scholar

show all references

##### References:
 [1] R. Aharoni and Y. Censor, Block-iterative projection methods for parallel computation of solutions to convex feasibility problems,, Linear Algebra Appl., 120 (1989), 165. doi: 10.1016/0024-3795(89)90375-3. Google Scholar [2] H. H. Bauschke, J. M. Borwein and P. L. Combettes, Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces,, Comm. Contemp. Math., 3 (2001), 615. doi: 10.1142/S0219199701000524. Google Scholar [3] H. H. Bauschke, J. M. Borwein and P. L. Combettes, Bregman monotone optimization algorithms,, SIAM J. Control Optim., 42 (2003), 596. doi: 10.1137/S0363012902407120. Google Scholar [4] H. H. Bauschke and P. L. Combettes, "Convex Analysis and Monotone Operator Theory in Hilbert Spaces,", Springer, (2011). doi: 10.1007/978-1-4419-9467-7. Google Scholar [5] H. H. Bauschke, E. Matoušková and S. Reich, Projection and proximal point methods: convergence results and counterexamples,, Nonlinear Anal., 56 (2004), 715. doi: 10.1016/j.na.2003.10.010. Google Scholar [6] V. Berinde, "Iterative Approximation of Fixed Points,", $2^{nd}$, (1912). Google Scholar [7] J. M. Borwein, S. Reich and S. Sabach, A characterization of Bregman firmly nonexpansive operators using a new monotonicity concept,, J. Nonlinear Convex Anal., 12 (2011), 161. Google Scholar [8] J. F. Bonnans and A. Shapiro, "Perturbation Analysis of Optimization Problems,", Springer, (2000). Google Scholar [9] L. M. Bregman, The method of successive projection for finding a common point of convex sets,, Soviet Math. Dokl., 6 (1965), 688. Google Scholar [10] L. M. Bregman, The relaxation method for finding a common point of convex sets and its application to the solution of problems in convex programming,, USSR Comput. Math. Math. Phys., 7 (1967), 200. Google Scholar [11] R. E. Bruck and S. Reich, Nonexpansive projections and resolvents of accretive operators in Banach spaces,, Houston J. Math., 3 (1977), 459. Google Scholar [12] D. Butnariu and Y. Censor, Strong convergence of almost simultaneous block-iterative projection methods in Hilbert spaces,, J. Comput. Appl. Math., 53 (1994), 33. doi: 10.1016/0377-0427(92)00123-Q. Google Scholar [13] D. Butnariu, Y. Censor and S. Reich, Iterative averaging of entropic projections for solving stochastic convex feasibility problems,, Comput. Optim. Appl., 8 (1997), 21. doi: 10.1023/A:1008654413997. Google Scholar [14] D. Butnariu and A. N. Iusem, "Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization,", Kluwer Academic Publishers, (2000). doi: 10.1007/978-94-011-4066-9. Google Scholar [15] D. Butnariu and E. Resmerita, Bregman distances, totally convex functions and a method for solving operator equations in Banach spaces,, Abstr. Appl. Anal., 2006 (2006), 1. doi: 10.1155/AAA/2006/84919. Google Scholar [16] Y. Censor, Row-action methods for huge and sparse systems and their applications,, SIAM Rev., 23 (1981), 444. doi: 10.1137/1023097. Google Scholar [17] Y. Censor and A. Lent, An iterative row-action method for interval convex programming,, J. Optim. Theory Appl., 34 (1981), 321. doi: 10.1007/BF00934676. Google Scholar [18] Y. Censor and S. Reich, Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization,, Optimization, 37 (1996), 323. doi: 10.1080/02331939608844225. Google Scholar [19] C. Chidume, "Geometric Properties of Banach Spaces and Nonlinear Iterations,", Lecture Notes in Mathematics 1965, (1965). Google Scholar [20] G. Cimmino, Calcolo approssimato per le soluzioni dei sistemi di equazioni lineari,, Ric. Sci. (Roma), 9 (1938), 326. Google Scholar [21] J. Dugundji and A. Granas, "Fixed Point Theory,", Springer, (2003). Google Scholar [22] A. Genel and J. Lindenstrauss, An example concerning fixed points,, Israel J. Math., 22 (1975), 81. Google Scholar [23] K. Goebel and S. Reich, "Uniform Convexity, Hyperbolic Ggeometry, and Nonexpansive Mappings,", Marcel Dekker, (1984). Google Scholar [24] S. Kaczmarz, Angenäherte Auflösung von Systemen linearer Gleichungen,, Bull. Internat. Acad. Polon. Sci. Lett. Sér. A Sci. Math., 35 (1937), 355. Google Scholar [25] G. Kassay, S. Reich and S. Sabach, Iterative methods for solving systems of variational inequalities in reflexive Banach spaces,, SIAM J. Optim., 21 (2011), 1319. doi: 10.1137/110820002. Google Scholar [26] M. Kikkawa and W. Takahashi, Approximating fixed points of nonexpansive mappings by the block iterative method in Banach spaces,, Int. J. Comput. Numer. Anal. Appl., 5 (2004), 59. Google Scholar [27] M. A. Krasnosel'ski, Two observations about the method of successive approximations,, Uspehi Math. Nauk., 10 (1955), 123. Google Scholar [28] M. A. Krasnosel'ski and P. P. Zabreiko, "Geometrical Methods of Nonlinear Analysis,", Springer, (1984). doi: 10.1007/978-3-642-69409-7. Google Scholar [29] W. R. Mann, Mean value methods in iteration,, Proc. Amer. Math. Soc., 4 (1953), 506. Google Scholar [30] S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces,, J. Math. Anal. Appl., 67 (1979), 274. doi: 10.1016/0022-247X(79)90024-6. Google Scholar [31] S. Reich, A weak convergence theorem for the alternating method with Bregman distances,, in, (1996), 313. Google Scholar [32] S. Reich and S. Sabach, A strong convergence theorem for a proximal-type algorithm in reflexive Banach spaces,, J. Nonlinear Convex Anal., 10 (2009), 471. Google Scholar [33] S. Reich and S. Sabach, Two strong convergence theorems for Bregman strongly nonexpansive operators in reflexive Banach spaces,, Nonlinear Analysis, 73 (2010), 122. doi: 10.1016/j.na.2010.03.005. Google Scholar [34] S. Reich and S. Sabach, Existence and approximation of fixed points of Bregman firmly nonexpansive mappings in reflexive Banach spaces,, in, (2011), 299. doi: 10.1007/978-1-4419-9569-8_15. Google Scholar [35] E. Zeidler, "Nonlinear Functional Analysis and Its Applications, Vol. I, Fixed-Point Theorems,", Springer, (1986). doi: 10.1007/978-1-4612-4838-5. Google Scholar
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