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Applications of fixed-point and optimization methods to the multiple-set split feasibility problem. (English) Zbl 1245.49051

Summary: The multiple-set split feasibility problem requires finding a point closest to a family of closed convex sets in one space such that its image under a linear transformation will be closest to another family of closed convex sets in the image space. It can be a model for many inverse problems where constraints are imposed on the solutions in the domain of a linear operator as well as in the operator’s range. It generalizes the convex feasibility problem as well as the two-set split feasibility problem. In this paper, we will review and report some recent results on iterative approaches to the multiple-set split feasibility problem.

MSC:

49N45 Inverse problems in optimal control
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[1] C. Byrne, “A unified treatment of some iterative algorithms in signal processing and image reconstruction,” Inverse Problems, vol. 20, no. 1, pp. 103-120, 2004. · Zbl 1051.65067
[2] H.-K. Xu, “Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces,” Inverse Problems, vol. 26, no. 10, p. 105018, 17, 2010. · Zbl 1213.65085
[3] Q. Yang, “The relaxed CQ algorithm solving the split feasibility problem,” Inverse Problems, vol. 20, no. 4, pp. 1261-1266, 2004. · Zbl 1066.65047
[4] C. Byrne, “Iterative oblique projection onto convex sets and the split feasibility problem,” Inverse Problems, vol. 18, no. 2, pp. 441-453, 2002. · Zbl 0996.65048
[5] Y. Censor, T. Elfving, N. Kopf, and T. Bortfeld, “The multiple-sets split feasibility problem and its applications for inverse problems,” Inverse Problems, vol. 21, no. 6, pp. 2071-2084, 2005. · Zbl 1089.65046
[6] B. Qu and N. Xiu, “A note on the CQ algorithm for the split feasibility problem,” Inverse Problems, vol. 21, no. 5, pp. 1655-1665, 2005. · Zbl 1080.65033
[7] J. Zhao and Q. Yang, “Self-adaptive projection methods for the multiple-sets split feasibility problem,” Inverse Problems, vol. 27, no. 3, Article ID 035009, 13 pages, 2011. · Zbl 1215.65115
[8] H.-K. Xu, “A variable Krasnosel’skii-Mann algorithm and the multiple-set split feasibility problem,” Inverse Problems, vol. 22, no. 6, pp. 2021-2034, 2006. · Zbl 1126.47057
[9] H.-K. Xu, “Averaged mappings and the gradient-projection algorithm,” Journal of Optimization Theory and Applications, vol. 150, no. 2, pp. 360-378, 2011. · Zbl 1233.90280
[10] Y. Dang and Y. Gao, “The strong convergence of a KM-CQ-like algorithm for a split feasibility problem,” Inverse Problems, vol. 27, no. 1, Article ID 015007, 9 pages, 2011. · Zbl 1211.65065
[11] F. Wang and H.-K. Xu, “Approximating curve and strong convergence of the CQ algorithm for the split feasibility problem,” Journal of Inequalities and Applications, vol. 2010, Article ID 102085, 13 pages, 2010. · Zbl 1189.65107
[12] Z. Wang, Q. Yang, and Y. Yang, “The relaxed inexact projection methods for the split feasibility problem,” Applied Mathematics and Computation, vol. 217, no. 12, pp. 5347-5359, 2011. · Zbl 1208.65088
[13] M. D. Altschuler and Y. Censor, “Feasibility solutions in radiation therapy treatment planning,” in Proceedings of the 8th International Conference on the Use of Computers in Radiation Therapy, pp. 220-224, IEEE Computer Society Press, Silver Spring, Md, USA, 1984.
[14] M. D. Altschuler, W.D. Powlis, and Y. Censor, “Teletherapy treatment planning with physician requirements included in the calculation: I. Concepts and methodology,” in Optimization of Cancer Radiotherapy, B. R. Paliwal, D. E. Herbert, and C. G. Orton, Eds., pp. 443-452, American Institute of Physics, New York, NY, USA, 1985.
[15] Y. Censor, “Mathematical aspects of radiation therapy treatment planning: continuous inversion versus full discretization and optimization versus feasibility,” in Computational Radiology and Imaging: Therapy and Diagnostics, C. Borgers and F. Natterer, Eds., vol. 110 of The IMA Volumes in Mathematics and Its Applications, pp. 101-112, Springer, New York, NY, USA, 1999. · Zbl 0929.65132
[16] Y. Censor, M. D. Altschuler, and W. D. Powlis, “A computational solution of the inverse problem in radiation-therapy treatment planning,” Applied Mathematics and Computation, vol. 25, no. 1, pp. 57-87, 1988. · Zbl 0637.65060
[17] F. Wang and H.-K. Xu, “Cyclic algorithms for split feasibility problems in Hilbert spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 12, pp. 4105-4111, 2011. · Zbl 1308.47079
[18] G. Lopez, V. Martin-Marquez, and H.-K. Xu, “Iterative algorithms for the multiple-sets split feasibility problem,” in Biomedical Mathematics: Promising Directions in Imaging, Therapy Planning and Inverse Problems, Y. Censor, M. Jiang, and G. Wang, Eds., pp. 243-279, Medical Physics Publishing, Madison, Wis, USA, 2009.
[19] Y. Censor and A. Segal, “The split common fixed point problem for directed operators,” Journal of Convex Analysis, vol. 16, no. 2, pp. 587-600, 2009. · Zbl 1189.65111
[20] Q. Yang and J. Zhao, “Generalized KM theorems and their applications,” Inverse Problems, vol. 22, no. 3, pp. 833-844, 2006. · Zbl 1117.65081
[21] J. Zhao and Q. Yang, “Several solution methods for the split feasibility problem,” Inverse Problems, vol. 21, no. 5, pp. 1791-1799, 2005. · Zbl 1080.65035
[22] Y. Censor, A. Motova, and A. Segal, “Perturbed projections and subgradient projections for the multiple-sets split feasibility problem,” Journal of Mathematical Analysis and Applications, vol. 327, no. 2, pp. 1244-1256, 2007. · Zbl 1253.90211
[23] W. Zhang, D. Han, and Z. Li, “A self-adaptive projection method for solving the multiple-sets split feasibility problem,” Inverse Problems, vol. 25, no. 11, Article ID 115001, 16 pages, 2009. · Zbl 1185.65102
[24] Y. Censor, T. Bortfeld, B. Martin, and A. Trofimov, “The split feasibility model leading to a unified approach for inversion problems in intensity-modulated radiation therapy,” Tech. Rep., Department of Mathematics, University of Haifa, Haifa, Israel, 2005.
[25] D. Han and W. Sun, “A new modified Goldstein-Levitin-Polyak projection method for variational inequality problems,” Computers & Mathematics with Applications, vol. 47, no. 12, pp. 1817-1825, 2004. · Zbl 1057.49011
[26] Y. Censor, T. Bortfeld, B. Martin, and A. Trofimov, “A unified approach for inversion problems in intensity-modulated radiation therapy,” Physics in Medicine and Biology, vol. 51, no. 10, pp. 2353-2365, 2006.
[27] E. K. Lee, T. Fox, and I. Crocker, “Integer programming applied to intensity-modulated radiation therapy treatment planning,” Annals of Operations Research, vol. 119, no. 1-4, pp. 165-181, 2003. · Zbl 1046.90050
[28] J. R. Palta and T. R. Mackie, Eds., Intensity-Modulated Radiation Therapy: The State of the Art, Medical Physical Monograph 29, American Association of Physists in Medicine, Medical Physical Publishing, Madison, Wis, USA, 2003.
[29] Q. Wu, R. Mohan, A. Niemierko, and R. Schmidt-Ullrich, “Optimization of intensity-modulated radiotherapy plans based on the equivalent uniform dose,” International Journal of Radiation Oncology Biology Physics, vol. 52, no. 1, pp. 224-235, 2002.
[30] B. Eicke, “Iteration methods for convexly constrained ill-posed problems in Hilbert space,” Numerical Functional Analysis and Optimization, vol. 13, no. 5-6, pp. 413-429, 1992. · Zbl 0769.65026
[31] E. S. Levitin and B. T. Poljak, “Minimization methods in the presence of constraints,” \vZurnal Vy\vcislitel’ noĭ Matematiki i Matemati Fiziki, vol. 6, pp. 787-823, 1966. · Zbl 0184.38902
[32] C. I. Podilchuk and R. J. Mammone, “Image recovery by convex projections using a least-squares constraint,” Journal of the Optical Society of America A, vol. 7, pp. 517-521, 1990.
[33] H. H. Bauschke and J. M. Borwein, “On projection algorithms for solving convex feasibility problems,” SIAM Review, vol. 38, no. 3, pp. 367-426, 1996. · Zbl 0865.47039
[34] M. Fukushima, “A relaxed projection method for variational inequalities,” Mathematical Programming, vol. 35, no. 1, pp. 58-70, 1986. · Zbl 0598.49024
[35] D. C. Youla, “On deterministic convergence of iterations of relaxed projection operators,” Journal of Visual Communication and Image Representation, vol. 1, no. 1, pp. 12-20, 1990.
[36] D. Youla, “Mathematical theory of image restoration by the method of convex projections,” in Image Recovery Theory and Applications, H. Stark, Ed., p. xx+543, Academic Press, Orlando, Fla, USA, 1987.
[37] M. I. Sezan and H. Stark, “Applications of convex projection theory to image recovery in tomography and related areas,” Image Recovery Theory and Applications, Academic Press, Orlando, Fla, USA, 1987. · Zbl 0627.65133
[38] A. Cegielski, “Generalized relaxation of nonexpansive operators and convex feasibility problems,” in Nonlinear Analysis and Optimization I. Nonlinear Analysis, vol. 513 of Contemporary Mathematics, pp. 111-123, American Mathematical Society, Providence, RI, USA, 2010. · Zbl 1217.47111
[39] C. Byrne, “Bregman-Legendre multidistance projection algorithms for convex feasibility and optimization,” in Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, vol. 8 of Studies in Computational Mathematics, pp. 87-99, North-Holland, Amsterdam, The Netherlands, 2001. · Zbl 0990.90094
[40] C. Byrne and Y. Censor, “Proximity function minimization using multiple Bregman projections, with applications to split feasibility and Kullback-Leibler distance minimization,” Annals of Operations Research, vol. 105, pp. 77-98, 2001. · Zbl 1012.90035
[41] Y. Censor, D. Gordon, and R. Gordon, “BICAV: A block-iterative parallel algorithm for sparse systems with pixel-related weighting,” IEEE Transactions on Medical Imaging, vol. 20, no. 10, pp. 1050-1060, 2001.
[42] Y. Censor, A. Gibali, and S. Reich, “The subgradient extragradient method for solving variational inequalities in Hilbert space,” Journal of Optimization Theory and Applications, vol. 148, no. 2, pp. 318-335, 2011. · Zbl 1229.58018
[43] Y. Censor and T. Elfving, “A multiprojection algorithm using Bregman projections in a product space,” Numerical Algorithms, vol. 8, no. 2-4, pp. 221-239, 1994. · Zbl 0828.65065
[44] J. M. Dye and S. Reich, “On the unrestricted iteration of projections in Hilbert space,” Journal of Mathematical Analysis and Applications, vol. 156, no. 1, pp. 101-119, 1991. · Zbl 0731.65041
[45] A. A. Goldstein, “Convex programming in Hilbert space,” Bulletin of the American Mathematical Society, vol. 70, pp. 709-710, 1964. · Zbl 0142.17101
[46] E. S. Levitin and B. T. Polyak, “Constrained minimization problems,” U.S.S.R. Computational Mathematics and Mathematical Physics., vol. 6, pp. 1-50, 1966. · Zbl 0161.07002
[47] B. S. He, H. Yang, Q. Meng, and D. R. Han, “Modified Goldstein-Levitin-Polyak projection method for asymmetric strongly monotone variational inequalities,” Journal of Optimization Theory and Applications, vol. 112, no. 1, pp. 129-143, 2002. · Zbl 0998.65066
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