A relaxed alternating CQ-algorithm for convex feasibility problems

https://doi.org/10.1016/j.na.2012.11.013Get rights and content

Abstract

Let H1,H2,H3 be real Hilbert spaces, let CH1, QH2 be two nonempty closed convex level sets, let A:H1H3, B:H2H3 be two bounded linear operators. Our interest is in solving the following new convex feasibility problem (1.1)Find xC,yQ such that Ax=By, which allows asymmetric and partial relations between the variables x and y. In this paper, we present and study the convergence of a relaxed alternating CQ-algorithm (RACQA) and show that the sequences generated by such an algorithm weakly converge to a solution of (1.1). The interest of RACQA is that we just need projections onto half-spaces, thus making the relaxed CQ-algorithm implementable. Note that, by taking B=I, in (1.1), we recover the split convex feasibility problem originally introduced in Censor and Elfving (1994) [13] and used later in intensity-modulated radiation therapy (Censor et al. (2006) [11]). We also recover the relaxed CQ-algorithm introduced by Yang (2004) [8] by particularizing both B and a given parameter.

Section snippets

Introduction and preliminaries

Due to their extraordinary utility and broad applicability in many areas of applied mathematics (most notably, fully-discretized models of problems in image reconstruction from projections, in image processing, and in intensity-modulated radiation therapy), algorithms for solving convex feasibility problems continue to receive great attention; see for instance [1], [2], [3], [4], [5] and also [6], [7], [8], [9]. In this paper our interest is in the study of the convergence of a relaxed

A weak convergence result

For sake of simplicity we set β=γ and remember that the Projection operator on a closed convex set K of a Hilbert space H admits the following useful characterization:

Given xH and zK, then z=PK(x) if, and only if, xz,yz0yK, and also has very attractive properties that make it particularly well suited for iterative algorithms. For instance, PK is firmly nonexpansive, namely for all x,yHPK(x)PK(y)2xy2(IPK)(x)(IPK)(y)2; see for instance Goebel and Reich [16].

Now, we are in a

Acknowledgments

I would like to thank N. Lehdili for his kind invitation to Jossigny and the two referees for their careful reading of the paper and for their pertinent comments and suggestions.

References (16)

  • A. Aleyner et al.

    Block-iterative algorithms for solving convex feasibility problems in Hilbert and in Banach

    Journal of Mathematical Analysis and Applications

    (2008)
  • H.-K. Xu

    Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces

    Inverse Problems

    (2010)
  • H.H. Bauschke et al.

    On projection algorithms for solving convex feasibility problems

    SIAM Review

    (1996)
  • C. Byrne

    A Unified Treatment of Some Iterative Algorithms in Signal Processing and Image Reconstruction

    (1984)
  • E. Masad et al.

    A note on the multiple-set split convex feasibility problem in Hilbert space

    Journal of Nonlinear and Convex Analysis

    (2007)
  • Y. Yao et al.

    Applications of fixed point and optimization methods to the multiple-sets split feasibility problem

    Journal of Applied Mathematics

    (2012)
  • H.K. Xu

    A variable Krasnosel’skii–Mann algorithm and the multiple-set split feasibility problem

    Inverse Problems

    (2006)
  • Q. Yang

    The relaxed CQ algorithm for solving the split feasibility problem

    Inverse Problems

    (2004)
There are more references available in the full text version of this article.

Cited by (139)

View all citing articles on Scopus
View full text