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A new halfspace-relaxation projection method for the split feasibility problem. (English) Zbl 1135.65022
Let $C$ and $Q$ be nonempty closed convex in $\Bbb R^{n}$ and $\Bbb R^{m}$, respectively, and $A$ an $m\times n$ real matrix. The problem, to find $ x\in C$ with $Ax\in Q$ if such $x$ exists, is called the split feasibility problem(SPF). The authors propose a new halfspace-relaxation projection method for the SFP. The method is implemented very easily and is proven to be fully convergent to the solution for the case where the solution set of the SFP is nonempty.

65F30Other matrix algorithms
Full Text: DOI
[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] Byrne, C. L.: Bregman -- Legendre mulidistance projection algorithms for convex feasibility and optimization, Inherently parallel algorithms in feasibility and optimization and their applications, 87-100 (2001) · Zbl 0990.90094
[3] Byrne, C. L.: Iterative oblique projection onto convex sets and the split feasibility problem, Inverse problems 18, 441-453 (2002) · Zbl 0996.65048 · doi:10.1088/0266-5611/18/2/310
[4] Byrne, C. L.: A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse problems 20, 103-120 (2004) · Zbl 1051.65067 · doi:10.1088/0266-5611/20/1/006
[5] Censor, Y.; Elfving, T.: A multiprojection algorithm using Bregman projections in a product space, Numer. algorithms 8, 221-239 (1994) · Zbl 0828.65065 · doi:10.1007/BF02142692
[6] Censor, Y.; Bortfeld, T.; Martin, B.; Trofimov, A.: A unified approach for inversion problems in intensity-modulated radiation therapy, Phys. med. Biol. 51, 2353-2365 (2006)
[7] Yang, Q. Z.: The relaxed CQ algorithm solving the split feasibility problem, Inverse problems 20, 1261-1266 (2004) · Zbl 1066.65047 · doi:10.1088/0266-5611/20/4/014
[8] Fukushima, M.: A relaxed projection method for variational inequalities, Math. program. 35, 58-70 (1986) · Zbl 0598.49024 · doi:10.1007/BF01589441
[9] Gafni, E. M.; Bertsekas, D. P.: Two-metric projection problems and descent methods for asymmetric variational inequality problems, Math. program. 53, 99-110 (1984)
[10] He, B. S.: A class of projection and contraction methods for monotone variational inequalities, Appl. math. Optim. 35, 69-76 (1997) · Zbl 0865.90119
[11] Polyak, B. T.: Minimization of unsmooth functionals, USSR comput. Math. math. Phys. 9, 14-29 (1969) · Zbl 0229.65056 · doi:10.1016/0041-5553(69)90061-5
[12] Qu, B.; Xiu, N. H.: A note on the CQ algorithm for the split feasibility problem, Inverse problems 21, 1655-1665 (2005) · Zbl 1080.65033 · doi:10.1088/0266-5611/21/5/009
[13] Rockafellar, R. T.: Convex analysis, (1970) · Zbl 0193.18401
[14] Solodov, M. V.; Tseng, P.: Modified projection-type methods for monotone variational inequalities, SIAM J. Control optimiz. 34, No. 5, 1814-1830 (1996) · Zbl 0866.49018 · doi:10.1137/S0363012994268655
[15] Toint, Ph.L.: Global convergence of a class of trust region methods for nonconvex minimization in Hilbert space, IMA J. Numer. anal. 8, 231-252 (1988) · Zbl 0698.65043 · doi:10.1093/imanum/8.2.231
[16] Zarantonello, E. H.: Projections on convex sets in Hilbert space and spectral theory, Contributions to nonlinear functional analysis (1971) · Zbl 0281.47043