zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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:
49N45Inverse problems in calculus of variations
WorldCat.org
Full Text: DOI
References:
[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 · doi:10.1088/0266-5611/20/1/006
[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 · doi:10.1088/0266-5611/26/10/105018
[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 · doi:10.1088/0266-5611/20/4/014
[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 · doi:10.1088/0266-5611/18/2/310
[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 · doi:10.1088/0266-5611/21/6/017
[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 · doi:10.1088/0266-5611/21/5/009
[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 · doi:10.1088/0266-5611/27/3/035009
[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 · doi:10.1088/0266-5611/22/6/007
[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 · doi:10.1007/s10957-011-9837-z
[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 · doi:10.1088/0266-5611/27/1/015007
[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 · doi:10.1155/2010/102085 · eudml:231154
[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 · doi:10.1016/j.amc.2010.11.058
[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 · doi:10.1016/0096-3003(88)90064-1
[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 · doi:10.1016/j.na.2011.03.044
[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 · http://www.heldermann.de/JCA/JCA16/JCA162/jca16031.htm
[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 · doi:10.1088/0266-5611/22/3/006
[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 · doi:10.1088/0266-5611/21/5/017
[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 · doi:10.1016/j.jmaa.2006.05.010
[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 · doi:10.1088/0266-5611/25/11/115001
[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 · doi:10.1016/j.camwa.2003.12.002
[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. · doi:10.1088/0031-9155/51/10/001
[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 · doi:10.1023/A:1022938707934
[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. · doi:10.1016/S0360-3016(01)02585-8
[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 · doi:10.1080/01630569208816489
[31] E. S. Levitin and B. T. Poljak, “Minimization methods in the presence of constraints,” \vZurnal Vy\vcislitel’ noĭ Matematiki i Matemati\vceskoĭ 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 · doi:10.1137/S0036144593251710
[34] M. Fukushima, “A relaxed projection method for variational inequalities,” Mathematical Programming, vol. 35, no. 1, pp. 58-70, 1986. · Zbl 0598.49024 · doi:10.1007/BF01589441
[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 · doi:10.1090/conm/513/10078
[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 · doi:10.1023/A:1013349430987
[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 · doi:10.1007/s10957-010-9757-3
[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 · doi:10.1007/BF02142692
[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 · doi:10.1016/0022-247X(91)90385-D
[45] A. A. Goldstein, “Convex programming in Hilbert space,” Bulletin of the American Mathematical Society, vol. 70, pp. 709-710, 1964. · Zbl 0142.17101 · doi:10.1090/S0002-9904-1964-11178-2
[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 · doi:10.1023/A:1013048729944