Relaxed extragradient methods for finding minimum-norm solutions of the split feasibility problem

Abstract In this paper, we consider the split feasibility problem (SFP) in infinite-dimensional Hilbert spaces, and study the relaxed extragradient methods for finding a common element of the solution set Γ of SFP and the set Fix ( S ) of fixed points of a nonexpansive mapping S . Combining Mann’s iterative method and Korpelevich’s extragradient method, we propose two iterative algorithms for finding an element of Fix ( S ) ∩ Γ . On one hand, for S = I , the identity mapping, we derive the strong convergence of one iterative algorithm to the minimum-norm solution of the SFP under appropriate conditions. On the other hand, we also derive the weak convergence of another iterative algorithm to an element of Fix ( S ) ∩ Γ under mild assumptions.

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

[2]  Z. Opial.  Weak convergence of the sequence of successive approximations for nonexpansive mappings , 1967 .

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

[4]  Qingzhi Yang,et al.  Several solution methods for the split feasibility problem , 2005 .

[5]  F. Browder,et al.  FIXED-POINT THEOREMS FOR NONCOMPACT MAPPINGS IN HILBERT SPACE. , 1965, Proceedings of the National Academy of Sciences of the United States of America.

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

[7]  Deren Han,et al.  Solving non-additive traffic assignment problems: A descent method for co-coercive variational inequalities , 2004, Eur. J. Oper. Res..

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

[9]  Y. Censor,et al.  A unified approach for inversion problems in intensity-modulated radiation therapy , 2006, Physics in medicine and biology.

[10]  Hong-Kun Xu.  Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces , 2010 .

[11]  Patrick L. Combettes,et al.  Signal Recovery by Proximal Forward-Backward Splitting , 2005, Multiscale Model. Simul..

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

[13]  Y. Censor,et al.  Perturbed projections and subgradient projections for the multiple-sets split feasibility problem , 2007 .

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

[15]  D. Bertsekas,et al.  Projection methods for variational inequalities with application to the traffic assignment problem , 1982 .

[16]  Y. Censor,et al.  The multiple-sets split feasibility problem and its applications for inverse problems , 2005 .

[17]  Wataru Takahashi,et al.  Weak Convergence Theorem by an Extragradient Method for Nonexpansive Mappings and Monotone Mappings , 2006 .

[18]  Yair Censor,et al.  Algorithms for the Split Variational Inequality Problem , 2010, Numerical Algorithms.

[19]  Hong-Kun Xu,et al.  AN IMPLICIT ITERATION PROCESS FOR NONEXPANSIVE MAPPINGS , 2001 .

[20]  P. L. Combettes,et al.  Solving monotone inclusions via compositions of nonexpansive averaged operators , 2004 .

[21]  Wataru Takahashi,et al.  Strong Convergence Theorem by a Hybrid Method for Nonexpansive Mappings and Lipschitz-Continuous Monotone Mappings , 2006, SIAM J. Optim..

[22]  L. Landweber.  An iteration formula for Fredholm integral equations of the first kind , 1951 .

[23]  Hong-Kun Xu,et al.  Convergence of Hybrid Steepest-Descent Methods for Variational Inequalities , 2003 .

[24]  Lu-Chuan Zeng,et al.  Implicit iteration scheme with perturbed mapping for common fixed points of a finite family of nonexpansive mappings , 2006 .

[25]  G. M. Korpelevich.  The extragradient method for finding saddle points and other problems , 1976 .

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