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Relaxed extragradient methods for finding minimum-norm solutions of the split feasibility problem. (English) Zbl 1236.47066
Summary: We consider the split feasibility problem (SFP) in infinite-dimensional Hilbert spaces, and study the relaxed extragradient methods for finding a common element of the solution set $\varGamma $ of SFP and the set $\text{Fix}(S)$ of fixed points of a nonexpansive mapping $S$. Combining Mann’s iterative method and Korpelevich’s extragradient method, we propose two iterative algorithms for finding an element of $\text{Fix}(S)\cap\varGamma$. On the one hand, for $S=I$, the identity mapping, we derive the strong convergence of one iterative algorithm to the minimum-norm solution of the SFP under appropriate conditions. On the other hand, we also derive the weak convergence of another iterative algorithm to an element of $\text{Fix}(S)\cap\varGamma$ under mild assumptions.

47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
65J20Improperly posed problems; regularization (numerical methods in abstract spaces)
65J22Inverse problems (numerical methods in abstract spaces)
Full Text: DOI
[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) · Zbl 1253.90211
[6] Censor, Y.; Segal, A.: Iterative projection methods in biomedical inverse problems, Mathematical methods in biomedical imaging and intensity-modulated therapy, 65-96 (2008)
[7] 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
[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] 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)
[10] 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
[11] 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
[12] 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
[13] 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
[14] 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
[15] Censor, Y.; Gibali, A.; Reich, S.: Algorithms for the split variational inequality problem, Numer. algorithms (2011) · Zbl 1232.58008
[16] 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
[17] 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
[18] Korpelevich, G. M.: An extragradient method for finding saddle points and for other problems, Ekonomika mat. Metody 12, 747-756 (1976) · Zbl 0342.90044
[19] 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
[20] Nadezhkina, N.; Takahashi, W.: Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz continuous monotone mappings, SIAM J. Optim. 16, 1230-1241 (2006) · Zbl 1143.47047 · doi:10.1137/050624315
[21] Geobel, K.; Kirk, W. A.: Topics in metric fixed point theory, Cambridge studies in advanced mathematics 28 (1990) · Zbl 0708.47031
[22] 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
[23] 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) · Zbl 1065.90015 · doi:10.1016/S0377-2217(03)00423-5
[24] 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
[25] Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. amer. Math. soc. 73, 595-597 (1967) · Zbl 0179.19902 · doi:10.1090/S0002-9904-1967-11761-0
[26] Browder, F. E.: Fixed point theorems for noncompact mappings in Hilbert spaces, Proc. natl. Acad. sci. USA 43, 1272-1276 (1965) · Zbl 0125.35801 · doi:10.1073/pnas.53.6.1272
[27] Xu, H. K.; Kim, T. H.: Convergence of hybrid steepest-descent methods for variational inequalities, J. optim. Theory appl. 119, 185-201 (2003) · Zbl 1045.49018 · doi:10.1023/B:JOTA.0000005048.79379.b6
[28] Xu, H. K.; Ori, R. G.: An implicit iteration process for nonexpansive mappings, Numer. funct. Anal. optim. 22, 767-773 (2001) · Zbl 0999.47043 · doi:10.1081/NFA-100105317
[29] Zeng, L. C.; Yao, J. C.: Implicit iteration scheme with perturbed mapping for common fixed points of a finite family of nonexpansive mappings, Nonlinear anal. 64, 2507-2515 (2006) · Zbl 1105.47061 · doi:10.1016/j.na.2005.08.028