An acceleration scheme for cyclic subgradient projections method. (English) Zbl 1267.90097
Summary: An algorithm for solving a convex feasibility problem for a finite family of convex sets is considered. The acceleration scheme of A. R. De Pierro [“em Methodos de projeção para a resolução de sistemas gerais de equações algébricas lineares”, Thesis (tese de Doutoramento), Instituto de Matemática da UFRJ, Cidade Universitária, Rio de Janeiro, Brasil (1981)], which is designed for simultaneous algorithms, is used in the algorithm to speed up the fully sequential cyclic subgradient projections method. A convergence proof is presented. The advantage of using this strategy is demonstrated with some examples.
MSC:
| 90C25 | Convex programming |
Software:
SNARK93References:
| [1] | Bauschke, H.H., Borwein, J.M.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38, 367–426 (1996) · Zbl 0865.47039 · doi:10.1137/S0036144593251710 |
| [2] | Butnariu, D., Censor, Y., Gurfil, P., Hadarn, E.: On the behavior of subgradient projections methods for convex feasibility problems in euclidean spaces. SIAM J. Optim. 19(2), 786–807 (2008) · Zbl 1161.49033 · doi:10.1137/070689127 |
| [3] | Cegielski, A.: Generalized relaxation of nonexpansive operators and convex feasibility problems. Contemp. Math. 513, 111–123 (2010) · Zbl 1217.47111 · doi:10.1090/conm/513/10078 |
| [4] | Cimmino, G.: Calcolo approssimato per le soluzioni dei sistemi di equazioni lineari. Ric. Sci. XVI, Ser. II, Anno IX 1, 326–333 (1938) · JFM 64.1244.02 |
| [5] | Censor, Y., Elfving, T., Herman, G.T., Nikazad, T.: On diagonally-relaxed orthogonal projection methods. SIAM J. Sci. Comput. 30, 473–504 (2007/08) · Zbl 1159.65317 · doi:10.1137/050639399 |
| [6] | Censor, Y., Lent, A.: Short communication: cyclic subgradient projections. Math. Program. 24(1), 233–235 (1982) · Zbl 0491.90077 · doi:10.1007/BF01585107 |
| [7] | Censor, Y.: Iterative methods for the convex feasibility problem. Ann. Discrete Math. 20, 83–91 (1984) · Zbl 0571.90068 |
| [8] | Combettes, P.L.: Convex set theoretic image recovery by extrapolated iterations of parallel subgradient projections. IEEE Trans. Image Process. 6, 493–506 (1997) · doi:10.1109/83.563316 |
| [9] | Browne, J.A., Herman, G.T., Odhner, D.: SNARK93: a programming system for image reconstruction from projections, the medical imaging processing group (MIPG). Technical report mipg198, Department of Radiology, University of Pennsylvania (1993) |
| [10] | De Pierro, A.R.: em Methodos de projeção para a resolução de sistemas gerais de equações algébricas lineares. Thesis (tese de Doutoramento), Instituto de Matemática da UFRJ, Cidade Universitária, Rio de Janeiro, Brasil (1981) |
| [11] | Dos Santos, L.T.: A parallel subgradient projections method for the convex feasibility problem. J. Comput. Appl. Math. 18, 307–320 (1987) · Zbl 0623.90076 · doi:10.1016/0377-0427(87)90004-5 |
| [12] | Evans, D.J., Popa, C.: Projections and preconditioning for inconsistent least-squares problems. Int. J. Comput. Math. 78(4), 599–616 (2001) · Zbl 1005.65037 · doi:10.1080/00207160108805134 |
| [13] | Gubin, L.G., Polyak, B.T., Raik, E.V.: The method of projections for finding the common point of convex sets. U.S.S.R. Comput. Math. Math. Phys. 7(6), 1–24 (1967) · Zbl 0199.51002 · doi:10.1016/0041-5553(67)90113-9 |
| [14] | Herman, G.T.: Fundamentals of Computerized Tomography: Image Reconstruction from Projections, 2nd edn. Springer, Berlin (2009) · Zbl 1280.92002 |
| [15] | Reich, S., Sabach, S.: Two strong convergence theorems for Bergman strongly nonexpansive operators in reflexive Banach spaces. Nonlinear Anal. 73(1), 122–135 (2010) · Zbl 1226.47089 · doi:10.1016/j.na.2010.03.005 |
| [16] | Santos, L.T.: Métodos de projeção do subgradiente para o problema de factibilidade convexa. Tese de Mestrado, IMECC-UNICAMP (1985) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
