zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Perturbed projections and subgradient projections for the multiple-sets split feasibility problem. (English) Zbl 1253.90211
Summary: We study the multiple-sets split feasibility problem that requires to find 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. By casting the problem into an equivalent problem in a suitable product space we are able to present a simultaneous subgradients projections algorithm that generates convergent sequences of iterates in the feasible case. We further derive and analyze a perturbed projection method for the multiple-sets split feasibility problem and, additionally, furnish alternative proofs to two known results.

90C30Nonlinear programming
65K10Optimization techniques (numerical methods)
90C47Minimax problems
Full Text: DOI
[1] Baillon, J. B.; Bruck, R. E.; Reich, S.: On the asymptotic behavior of nonexpansive mappings and semigroups in Banach spaces. Houston J. Math. 4, 1-9 (1978) · Zbl 0396.47033
[2] Bauschke, H. H.; Borwein, J. M.: On projection algorithms for solving convex feasibility problems. SIAM rev. 38, 367-426 (1996) · Zbl 0865.47039
[3] Bauschke, H. H.; Combettes, P. L.; Kruk, S. G.: Extrapolation algorithm for affine-convex feasibility problems. Numer. algorithms 41, 239-274 (2006) · Zbl 1098.65060
[4] Bruck, R. E.; Reich, S.: Nonexpansive projections and resolvents of accretive operators in Banach spaces. Houston J. Math. 3, 459-470 (1977) · Zbl 0383.47035
[5] Byrne, C.: Bregman -- Legendre multidistance projection algorithms for convex feasibility and optimization. Inherently parallel algorithms in feasibility and optimization and their applications, 87-100 (2001) · Zbl 0990.90094
[6] Byrne, C.: Iterative oblique projection onto convex sets and the split feasibility problem. Inverse problems 18, 441-453 (2002) · Zbl 0996.65048
[7] Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse problems 20, 103-120 (2004) · Zbl 1051.65067
[8] Byrne, C.; Censor, Y.: Proximity function minimization using multiple Bregman projections, with applications to split feasibility and Kullback -- Leibler distance minimization. Ann. oper. Res. 105, 77-98 (2001) · Zbl 1012.90035
[9] A. Cegielski, Convergence of the projected surrogate constraints method for the linear split feasibility problems, J. Convex Anal. 14 (2007), in press · Zbl 1128.65039
[10] A. Cegielski, Projection methods for the linear split feasibility problems, Technical Report, September 14, 2004, Institute of Mathematics, University of Zielona Góra, Zielona Góra, Poland · Zbl 1148.65037
[11] Censor, Y.: Row-action methods for huge and sparse systems and their applications. SIAM rev. 23, 444-466 (1981) · Zbl 0469.65037
[12] Censor, Y.; Bortfeld, T.; Martin, B.; Trofimov, A.: A unified approach for inversion problems in intensity-modulated radiation therapy. Phys. medicine biology 51, 2353-2365 (2006)
[13] Censor, Y.; Elfving, T.: A multiprojection algorithm using Bregman projections in a product space. Numer. algorithms 8, 221-239 (1994) · Zbl 0828.65065
[14] 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
[15] Censor, Y.; Gordon, D.; Gordon, R.: BICAV: A block-iterative, parallel algorithm for sparse systems with pixel-related weighting. IEEE trans. Medical imaging 20, 1050-1060 (2001)
[16] Censor, Y.; Lent, A.: Cyclic subgradient projections. Math. programming 24, 233-235 (1982) · Zbl 0491.90077
[17] Censor, Y.; Zenios, S. A.: Parallel optimization: theory, algorithms, and applications. (1997) · Zbl 0945.90064
[18] Crombez, G.: Non-monotoneous parallel iteration for solving convex feasibility problems. Kybernetika 39, 547-560 (2003) · Zbl 1249.65040
[19] G. Crombez, A sequential iteration algorithm with non-monotoneous behaviour in the method of projections onto convex sets, Czechoslovak Math. J., in press · Zbl 1164.47399
[20] Golshtein, E.; Tretyakov, N.: Modified Lagrangians and monotone maps in optimization. (1996) · Zbl 0848.49001
[21] Noor, M. Aslam: Some developments in general variational inequalities. Appl. math. Comput. 152, 197-277 (2004) · Zbl 1134.49304
[22] Pierra, G.: Decomposition through formalization in a product space. Math. programming 28, 96-115 (1984) · Zbl 0523.49022
[23] Qu, B.; Xiu, N.: A note on the CQ algorithm for the split feasibility problem. Inverse problems 21, 1655-1665 (2005) · Zbl 1080.65033
[24] Reich, S.: Averaged mappings in the Hilbert ball. J. math. Anal. appl. 109, 199-206 (1985) · Zbl 0588.47061
[25] P.S.M. Santos and S. Scheimberg, A projection algorithm for general variational inequalities with perturbed constraint sets, Appl. Math. Comput. (2006), in press · Zbl 1148.65308
[26] Stark, H.; Yang, Y.: Vector space projections: A numerical approach to signal and image processing, neural nets, and optics. (1998) · Zbl 0903.65001
[27] Yamada, I.: Hybrid steepest descent method for variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings. Numer. funct. Anal. optim. 25, 619-655 (2004) · Zbl 1095.47049
[28] Yang, Q.: The relaxed CQ algorithm solving the split feasibility problem. Inverse problems 20, 1261-1266 (2004) · Zbl 1066.65047
[29] Yang, Q.: On variable-step relaxed projection algorithm for variational inequalities. J. math. Anal. appl. 302, 166-179 (2005) · Zbl 1056.49018
[30] Yang, Q.; Zhao, J.: Generalized KM theorems and their applications. Inverse problems 22, 833-844 (2006) · Zbl 1117.65081
[31] Zhao, J.; Yang, Q.: Several solution methods for the split feasibility problem. Inverse problems 21, 1791-1799 (2005) · Zbl 1080.65035