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“Optimal” choice of the step length of the projection and contraction methods for solving the split feasibility problem. (English) Zbl 1518.47102

Summary: In this paper, first, we review the projection and contraction methods for solving the split feasibility problem (SFP), and then by using the inverse strongly monotone property of the underlying operator of the SFP, we improve the “optimal” step length to provide the modified projection and contraction methods. Also, we consider the corresponding relaxed variants for the modified projection and contraction methods, where the two closed convex sets are both level sets of convex functions. Some convergence theorems of the proposed methods are established under suitable conditions. Finally, we give some numerical examples to illustrate that the modified projection and contraction methods have an advantage over other methods, and improve greatly the projection and contraction methods.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H05 Monotone operators and generalizations
47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
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[1] Antipin, AS, On a method for convex programs using a symmetrical modification of the Lagrange function, Ekon. Mat. Metody, 12, 1164-1173, (1976) · Zbl 0368.90115
[2] Bauschke, H.H., Combettes, P.L.: Convex Analysis and Motonone Operator Theory in Hilbert Spaces. Springer, London (2011) · Zbl 1218.47001 · doi:10.1007/978-1-4419-9467-7
[3] Bauschke, HH; Borwein, JM, On projection algorithms for solving convex feasibility problems, SIAM Rev., 38, 367-426, (1996) · Zbl 0865.47039 · doi:10.1137/S0036144593251710
[4] Bnouhachem, A; Noor, MA; Khalfaoui, M; Zhaohan, S, On descent-projection method for solving the split feasibility problems, J. Glob. Optim., 54, 627-639, (2012) · Zbl 1282.90190 · doi:10.1007/s10898-011-9782-2
[5] Byrne, CL, Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Probl., 18, 441-453, (2002) · Zbl 0996.65048 · doi:10.1088/0266-5611/18/2/310
[6] Byrne, CL, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Probl., 20, 103-120, (2004) · Zbl 1051.65067 · doi:10.1088/0266-5611/20/1/006
[7] Cai, X; Gu, G; He, B, On the O(1/t) convergence rate of the projection and contraction methods for variational inequalities with Lipschitz continuous monotone operators, Comput. Optim. Appl., 57, 339-363, (2014) · Zbl 1304.90203 · doi:10.1007/s10589-013-9599-7
[8] Censor, Y; Bortfeld, T; Martin, B; Trofimov, A, A unified approach for inversion problems in intensitymodulated radiation therapy, Phys. Med. Biol., 51, 2353-2365, (2006) · doi:10.1088/0031-9155/51/10/001
[9] 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
[10] Censor, Y; Elfving, T; Kopf, N; Bortfeld, T, The multiple-sets split feasibility problem and its applications for inverse problems, Inverse Probl., 21, 2071-2084, (2005) · Zbl 1089.65046 · doi:10.1088/0266-5611/21/6/017
[11] Censor, Y; Gibali, A; Reich, S, The subgradient extragradient method for solving variational inequalities in Hilbert space, J. Optim. Theory Appl., 148, 318-335, (2011) · Zbl 1229.58018 · doi:10.1007/s10957-010-9757-3
[12] Dang, Y; Gao, Y, The strong convergence of a KM-CQ-like algorithm for a split feasibility problem, Inverse Probl., 27, 015007, (2011) · Zbl 1211.65065 · doi:10.1088/0266-5611/27/1/015007
[13] Dong, QL; Lu, YY; Yang, J, The extragradient algorithm with inertial effects for solving the variational inequality, Optim., 65, 2217-2226, (2016) · Zbl 1358.90139 · doi:10.1080/02331934.2016.1239266
[14] Dong, QL; Yao, Y; He, S, Weak convergence theorems of the modified relaxed projection algorithms for the split feasibility problem in Hilbert spaces, Optim. Lett., 8, 1031-1046, (2014) · Zbl 1320.90103 · doi:10.1007/s11590-013-0619-4
[15] Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, Berlin (2003) · Zbl 1062.90001
[16] Fukushima, MA, Relaxed projection method for variational inequalities, Math. Program., 35, 58-70, (1986) · Zbl 0598.49024 · doi:10.1007/BF01589441
[17] Gibali, A., Liu, L., Tang, Y.C.: Note on the modified relaxation CQ algorithm for the split feasibility problem. Optim. Lett. (2017). https://doi.org/10.1007/s11590-017-1148-3 · Zbl 1423.90179
[18] He, BS, A class of projection and contraction methods for monotone variational inequalities, Appl. Math. Optim., 35, 69-76, (1997) · Zbl 0865.90119 · doi:10.1007/BF02683320
[19] Korpelevich, GM, The extragradient method for finding saddle points and other problems, Ekon. Mate. Metody, 12, 747-756, (1976) · Zbl 0342.90044
[20] Latif, A; Qin, X, A regularization algorithm for a splitting feasibility problem in Hilbert spaces, J. Nonlinear Sci. Appl., 10, 3856-3862, (2017) · Zbl 1412.47034 · doi:10.22436/jnsa.010.07.40
[21] Latif, A; Vahidi, J; Eslamian, M, Strong convergence for generalized multiple-set split feasibility problem, Filomat, 30, 459-467, (2016) · Zbl 1464.47037 · doi:10.2298/FIL1602459L
[22] López, G; Martín-Márquez, V; Wang, F; Xu, HK, Solving the split feasibility problem without prior knowledge of matrix norms, Inverse Probl., 27, 085004, (2012) · Zbl 1262.90193 · doi:10.1088/0266-5611/28/8/085004
[23] Qu, B; Xiu, N, A new halfspace-relaxation projection method for the split feasibility problem, Linear Algebra Appl., 428, 1218-1229, (2008) · Zbl 1135.65022 · doi:10.1016/j.laa.2007.03.002
[24] Qu, B; Xiu, N, A note on the CQ algorithm for the split feasibility problem, Inverse Probl., 21, 1655-1665, (2005) · Zbl 1080.65033 · doi:10.1088/0266-5611/21/5/009
[25] Rudin, W.: Functional Analysis, 2nd edn. McGraw-Hill, New York (1991) · Zbl 0867.46001
[26] Suantai, S; Cholamjiak, P; Cho, YJ; Cholamjiak, W, On solving split equilibrium problems and fixed point problems of nonspreading multi-valued mappings in Hilbert spaces, Fixed Point Theory Appl., 2016, 35, (2016) · Zbl 1346.47061 · doi:10.1186/s13663-016-0509-4
[27] Sun, DF, A class of iterative methods for solving nonlinear projection equations, J. Optim. Theory Appl., 91, 123-140, (1996) · Zbl 0871.90091 · doi:10.1007/BF02192286
[28] Tang, Y; Zhu, C; Yu, H, Iterative methods for solving the multiple-sets split feasibility problem with splitting self-adaptive step size, Fixed Point Theory Appl., 2015, 178, (2015) · Zbl 1338.90309 · doi:10.1186/s13663-015-0430-2
[29] Tibshirani, R, Regression shrinkage and selection via the lasso, J. R. Stat. Soc. Ser. B. Stat. Methodol., 58, 267-288, (1996) · Zbl 0850.62538
[30] Tseng, P, A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control Optim., 38, 431-446, (2000) · Zbl 0997.90062 · doi:10.1137/S0363012998338806
[31] Wang, F, Polyak’s gradient method for split feasibility problem constrained by level sets, Numer. Algor., 77, 925-938, (2018) · Zbl 1486.65054 · doi:10.1007/s11075-017-0347-4
[32] Wang, Z; Yang, Q; Yang, Y, The relaxed inexact projection methods for the split feasibility problem, Appl. Math. Comput., 217, 5347-5359, (2011) · Zbl 1208.65088
[33] Wen, M; Peng, J; Tang, YC, A cyclic and simultaneous iterative method for solving the multiple-sets split feasibility problem, J. Optim. Theory Appl., 166, 844-860, (2015) · Zbl 1330.90081 · doi:10.1007/s10957-014-0701-9
[34] Xu, HK, Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces, Inverse Probl., 26, 105018, (2010) · Zbl 1213.65085 · doi:10.1088/0266-5611/26/10/105018
[35] Yang, Q, On variable-step relaxed projection algorithm for variational inequalities, J. Math. Anal. Appl., 302, 166-179, (2005) · Zbl 1056.49018 · doi:10.1016/j.jmaa.2004.07.048
[36] Yao, Y; Yao, Z; Abdou, AAN; Cho, YJ, Self-adaptive algorithms for proximal split feasibility problems and strong convergence analysis, Fixed Point Theory Appl., 2015, 205, (2015) · Zbl 1346.49052 · doi:10.1186/s13663-015-0462-7
[37] Yen, LH; Muu, LD; Huyen, NTT, An algorithm for a class of split feasibility problems: application to a model in electricity production, Math. Methods Oper. Res., 84, 549-565, (2016) · Zbl 1370.90277 · doi:10.1007/s00186-016-0553-1
[38] Zhang, W; Han, D; Li, Z, A self-adaptive projection method for solving the multiple-sets split feasibility problem, Inverse Probl., 25, 115001, (2009) · Zbl 1185.65102 · doi:10.1088/0266-5611/25/11/115001
[39] Zhao, J; Yang, Q, A simple projection method for solving the multiple-sets split feasibility problem, Inverse Probl. Sci. Eng., 21, 537-546, (2013) · Zbl 1285.65038 · doi:10.1080/17415977.2012.712521
[40] Zhao, J; Zhang, Y; Yang, Q, Modified projection methods for the split feasibility problem and the multiple-sets split feasibility problem, Appl. Math. Comput., 219, 1644-1653, (2012) · Zbl 1291.90179
[41] Zhao, J; Yang, Q, Self-adaptive projection methods for the multiple-sets split feasibility problem, Inverse Probl., 27, 035009, (2011) · Zbl 1215.65115 · doi:10.1088/0266-5611/27/3/035009
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