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An extragradient method for solving split feasibility and fixed point problems. (English) Zbl 1252.65102
Summary: The purpose of this paper is to introduce and analyze an extragradient method with regularization for finding a common element of the solution set Γ of the split feasibility problem and the set Fix (S) of fixed points of a nonexpansive mapping S in the setting of infinite-dimensional Hilbert spaces. Combining the regularization method and the extragradient method due to N. Nadezhkina and W. Takahashi [J. Optim. Theory Appl. 128, No. 1, 191–201 (2006; Zbl 1130.90055)], we propose an iterative algorithm for finding an element of Fix (S)Γ. We prove that the sequences generated by the proposed algorithm converge weakly to an element of Fix (S)Γ under mild conditions.

65K10Optimization techniques (numerical methods)
47J25Iterative procedures (nonlinear operator equations)
[1]Censor, Y.; Elfving, T.: A multiprojection algorithm using Bregman projections in a product space, Numer. algorithms 8, 221-239 (1994) · Zbl 0828.65065 · doi:10.1007/BF02142692
[2]Byrne, C.: Iterative oblique projection onto convex subsets and the split feasibility problem, Inverse problems 18, 441-453 (2002) · Zbl 0996.65048 · doi:10.1088/0266-5611/18/2/310
[3]Censor, Y.; Bortfeld, T.; Martin, B.; Trofimov, A.: A unified approach for inversion problems in intensity-modulated radiation therapy, Phys. med. Biol. 51, 2353-2365 (2006)
[4]Censor, Y.; Elfving, T.; Kopf, N.; Bortfeld, T.: The multiple-sets split feasibility problem and its applications for inverse problems, Inverse problems 21, 2071-2084 (2005) · Zbl 1089.65046 · doi:10.1088/0266-5611/21/6/017
[5]Censor, Y.; Motova, A.; Segal, A.: Perturbed projections and subgradient projections for the multiple-sets split feasibility problem, J. math. Anal. appl. 327, 1244-1256 (2007)
[6]Xu, H. K.: Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces, Inverse problems 26, 105018 (2010) · Zbl 1213.65085 · doi:10.1088/0266-5611/26/10/105018
[7]Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse problems 20, 103-120 (2004) · Zbl 1051.65067 · doi:10.1088/0266-5611/20/1/006
[8]Qu, B.; Xiu, N.: A note on the CQ algorithm for the split feasibility problem, Inverse problems 21, 1655-1665 (2005) · Zbl 1080.65033 · doi:10.1088/0266-5611/21/5/009
[9]Xu, H. K.: A variable Krasnoselskii–Mann algorithm and the multiple-set split feasibility problem, Inverse problems 22, 2021-2034 (2006) · Zbl 1126.47057 · doi:10.1088/0266-5611/22/6/007
[10]Yang, Q.: The relaxed CQ algorithm for solving the split feasibility problem, Inverse problems 20, 1261-1266 (2004) · Zbl 1066.65047 · doi:10.1088/0266-5611/20/4/014
[11]Zhao, J.; Yang, Q.: Several solution methods for the split feasibility problem, Inverse problems 21, 1791-1799 (2005) · Zbl 1080.65035 · doi:10.1088/0266-5611/21/5/017
[12]Sezan, M. I.; Stark, H.: Applications of convex projection theory to image recovery in tomography and related areas, Image recovery theory and applications, 415-462 (1987)
[13]Eicke, B.: Iteration methods for convexly constrained ill-posed problems in Hilbert spaces, Numer. funct. Anal. optim. 13, 413-429 (1992) · Zbl 0769.65026 · doi:10.1080/01630569208816489
[14]Landweber, L.: An iterative formula for Fredholm integral equations of the first kind, Amer. J. Math. 73, 615-624 (1951) · Zbl 0043.10602 · doi:10.2307/2372313
[15]Potter, L. C.; Arun, K. S.: A dual approach to linear inverse problems with convex constraints, SIAM J. Control optim. 31, 1080-1092 (1993) · Zbl 0797.49019 · doi:10.1137/0331049
[16]Combettes, P. L.; Wajs, V.: Signal recovery by proximal forward–backward splitting, Multiscale model. Simul. 4, 1168-1200 (2005) · Zbl 1179.94031 · doi:10.1137/050626090
[17]Korpelevich, G. M.: An extragradient method for finding saddle points and for other problems, Ekonomika mat. Metody 12, 747-756 (1976)
[18]Nadezhkina, N.; Takahashi, W.: Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, J. optim. Theory appl. 128, 191-201 (2006) · Zbl 1130.90055 · doi:10.1007/s10957-005-7564-z
[19]Kinderlehrar; Stampacchia, G.: An introduction to variational inequalities and their applications, (1980)
[20]Bertsekas, D. P.; Gafni, E. M.: Projection methods for variational inequalities with applications to the traffic assignment problem, Math. program. Stud. 17, 139-159 (1982) · Zbl 0478.90071
[21]Han, D.; Lo, H. K.: Solving non-additive traffic assignment problems: a descent method for co-coercive variational inequalities, European J. Oper. res. 159, 529-544 (2004)
[22]Combettes, P. L.: Solving monotone inclusions via compositions of nonexpansive averaged operators, Optimization 53, No. 5–6, 475-504 (2004) · Zbl 1153.47305 · doi:10.1080/02331930412331327157
[23]Geobel, K.; Kirk, W. A.: Topics in metric fixed point theory, Cambridge studies in advanced mathematics 28 (1990) · Zbl 0708.47031
[24]Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. amer. Math. soc. 73, 591-597 (1967) · Zbl 0179.19902 · doi:10.1090/S0002-9904-1967-11761-0
[25]Tan, K. K.; Xu, H. K.: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. math. Anal. appl. 178, 301-308 (1993) · Zbl 0895.47048 · doi:10.1006/jmaa.1993.1309
[26]Osilike, M. O.; Aniagbosor, S. C.; Akuchu, B. G.: Fixed points of asymptotically demicontractive mappings in arbitrary Banach space, Panamer. math. J. 12, 77-88 (2002) · Zbl 1018.47047
[27]Rockafellar, R. T.: On the maximality of sums of nonlinear monotone operators, Trans. amer. Math. soc. 149, 75-88 (1970) · Zbl 0222.47017 · doi:10.2307/1995660