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Keywords:
multiple-sets split feasibility problem; subgradient; extrapolated technique
multiple-sets split feasibility problem; subgradient; extrapolated technique
Summary:
In this paper, we present a simultaneous subgradient algorithm for solving the multiple-sets split feasibility problem. The algorithm employs two extrapolated factors in each iteration, which not only improves feasibility by eliminating the need to compute the Lipschitz constant, but also enhances flexibility due to applying variable step size. The convergence of the algorithm is proved under suitable conditions. Numerical results illustrate that the new algorithm has better convergence than the existing one.
References:
[1] Bauschke, H. H., Borwein, J. M.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38 (1996), 367-426. DOI 10.1137/S0036144593251710 | MR 1409591 | Zbl 0865.47039
[2] Byrne, C.: Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Probl. 18 (2002), 441-453. MR 1910248 | Zbl 0996.65048
[3] Censor, Y., Bortfeld, T., Martin, B., Trofimov, A.: A unified approach for inversion problems in intensity-modulated radiation therapy. Physics in Medicine and Biology 51 (2006), 2353-2365.
[4] Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms 8 (1994), 221-239. DOI 10.1007/BF02142692 | MR 1309222 | Zbl 0828.65065
[5] Censor, Y., Elfving, T., Kopf, N., Bortfeld, T.: The multiple-sets split feasibility problem and its applications for inverse problems. Inverse Probl. 21 (2005), 2071-2084. MR 2183668 | Zbl 1089.65046
[6] Censor, Y., Motova, A., Segal, A.: Perturbed projections and subgradient projections for the multiple-sets split feasibility problem. J. Math. Anal. Appl. 327 (2007), 1244-1256. DOI 10.1016/j.jmaa.2006.05.010 | MR 2280001 | Zbl 1253.90211
[7] Censor, Y., Segal, A.: Sparse string-averaging and split common fixed points. Nonlinear Analysis and Optimization I. Nonlinear Analysis. A conference in celebration of Alex Ioffe's 70th and Simeon Reich's 60th birthdays, Haifa, Israel, June 18-24, 2008 A. Leizarowitz et al. Contemporary Mathematics 513 American Mathematical Society, Providence (2010), 125-142. MR 2668242 | Zbl 1229.47107
[8] Censor, Y., Segal, A.: The split common fixed point problem for directed operators. J. Convex Anal. 16 (2009), 587-600. MR 2559961 | Zbl 1189.65111
[9] Combettes, P. L.: Convex set theoretic image reconvery by extrapolated iterations of parallel subgradient projections. IEEE Transactions on Image Processing 6 (1997), 493-506. DOI 10.1109/83.563316
[10] Dang, Y., Gao, Y.: Non-monotonous accelerated parallel subgradient projection algorithm for convex feasibility problem. Optimization (electronic only) (2012). MR 3195995
[11] Dang, Y., Gao, Y.: The strong convergence of a KM-CQ-like algorithm for a split feasibility problem. Inverse Probl. 27 (2011), Article ID 015007. MR 2746410 | Zbl 1211.65065
[12] Masad, E., Reich, S.: A note on the multiple-set split convex feasibility problem in Hilbert space. J. Nonlinear Convex Anal. 8 (2007), 367-371. MR 2377859 | Zbl 1171.90009
[13] Pierra, G.: Decomposition through formalization in a product space. Math. Program. 28 (1984), 96-115. DOI 10.1007/BF02612715 | MR 0727421 | Zbl 0523.49022
[14] Pierra, G.: Parallel constraint decomposition for minimization of a quadratic form. Optimization Techniques. Modeling and Optimization in the Service of Man Part 2. Proceedings, 7th IFIP conference, Nice, September 8-12, 1975 J. Cea Lecture Notes in Computer Science 41 Springer, Berlin (1976), 200-218 French.
[15] Xu, H.-K.: A variable Krasnosel'skiĭ-Mann algorithm and the multiple-set split feasibility problem. Inverse Probl. 22 (2006), 2021-2034. MR 2277527 | Zbl 1126.47057
[16] Yang, Q.: The relaxed CQ algorithm solving the split feasibility problem. Inverse Probl. 20 (2004), 1261-1266. MR 2087989 | Zbl 1066.65047
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