Approximating Curve and Strong Convergence of the CQ Algorithm for the Split Feasibility Problem

Using the idea of Tikhonov's regularization, we present properties of the approximating curve for the split feasibility problem (SFP) and obtain the minimum-norm solution of SFP as the strong limit of the approximating curve. It is known that in the infinite-dimensional setting, Byrne's CQ algorithm (Byrne, 2002) has only weak convergence. We introduce a modification of Byrne's CQ algorithm in such a way that strong convergence is guaranteed and the limit is also the minimum-norm solution of SFP.

[1]  C. Byrne,et al.  A unified treatment of some iterative algorithms in signal processing and image reconstruction , 2003 .

[2]  N. Lloyd.  TOPICS IN METRIC FIXED POINT THEORY (Cambridge Studies in Advanced Mathematics 28) , 1992 .

[3]  M. Birkner,et al.  Blow-up of semilinear PDE's at the critical dimension. A probabilistic approach , 2002 .

[4]  J. Aubin.  Optima and Equilibria: An Introduction to Nonlinear Analysis , 1993 .

[5]  Y. Alber,et al.  Nonlinear Ill-posed Problems of Monotone Type , 2006 .

[6]  Yair Censor,et al.  A multiprojection algorithm using Bregman projections in a product space , 1994, Numerical Algorithms.

[7]  H. Engl,et al.  Regularization of Inverse Problems , 1996 .

[8]  W. R. Mann,et al.  Mean value methods in iteration , 1953 .

[9]  Qingzhi Yang.  The relaxed CQ algorithm solving the split feasibility problem , 2004 .

[10]  Hong-Kun Xu.  Iterative Algorithms for Nonlinear Operators , 2002 .

[11]  N. Xiu,et al.  A note on the CQ algorithm for the split feasibility problem , 2005 .

[12]  C. Byrne,et al.  Iterative oblique projection onto convex sets and the split feasibility problem , 2002 .

[13]  W. A. Kirk,et al.  Topics in Metric Fixed Point Theory , 1990 .

[14]  Bertolt Eicke.  Iteration methods for convexly constrained ill-posed problems in hilbert space , 1992 .

[15]  Hong-Kun Xu.  A variable Krasnosel'skii–Mann algorithm and the multiple-set split feasibility problem , 2006 .

[16]  Qingzhi Yang,et al.  Generalized KM theorems and their applications , 2006 .

[17]  J. Lindenstrauss,et al.  An example concerning fixed points , 1975 .

[18]  Andreas Neubauer,et al.  Tikhonov-regularization of ill-posed linear operator equations on closed convex sets , 1988 .

[19]  Osman Güer.  On the convergence of the proximal point algorithm for convex minimization , 1991 .