Generalized relaxation of string averaging operators based on strictly relaxed cutter operators

Abstract. We present convergence analysis of a generalized relaxation of string averaging operators which is based on strictly relaxed cutter operators on a general Hilbert space. In this paper, the string averaging operator is assembled by averaging of strings’ endpoints and each string consists of composition of finitely many strictly relaxed cutter operators. We also consider projected version of the generalized relaxation of string averaging operator. To evaluate the study, we recall a wide class of iterative methods for solving linear equations (inequalities) and use the subgradient projection method for solving nonlinear convex feasibility problems.

[1]  S. Reich,et al.  A modular string averaging procedure for solving the common fixed point problem for quasi-nonexpansive mappings in Hilbert space , 2016, Numerical Algorithms.

[2]  Andrzej Cegielski,et al.  General Method for Solving the Split Common Fixed Point Problem , 2015, J. Optim. Theory Appl..

[3]  Touraj Nikazad,et al.  Perturbation-Resilient Iterative Methods with an Infinite Pool of Mappings , 2015, SIAM J. Numer. Anal..

[4]  Yair Censor,et al.  Strict Fejér Monotonicity by Superiorization of Feasibility-Seeking Projection Methods , 2014, J. Optim. Theory Appl..

[5]  Yair Censor,et al.  String-averaging projected subgradient methods for constrained minimization , 2013, Optim. Methods Softw..

[6]  A. Cegielski.  Iterative Methods for Fixed Point Problems in Hilbert Spaces , 2012 .

[7]  Yair Censor,et al.  Convergence and perturbation resilience of dynamic string-averaging projection methods , 2012, Computational Optimization and Applications.

[8]  Andrzej Cegielski,et al.  Extrapolation and Local Acceleration of an Iterative Process for Common Fixed Point Problems , 2012, 1204.4284.

[9]  G. Herman,et al.  Accelerated perturbation-resilient block-iterative projection methods with application to image reconstruction , 2012, Inverse problems.

[10]  T. Elfving,et al.  Properties of a class of block-iterative methods , 2009 .

[11]  Ran Davidi,et al.  Perturbation-resilient block-iterative projection methods with application to image reconstruction from projections , 2009, Int. Trans. Oper. Res..

[12]  Yair Censor,et al.  On the string averaging method for sparse common fixed-point problems , 2009, Int. Trans. Oper. Res..

[13]  Simeon Reich,et al.  Block-iterative algorithms for solving convex feasibility problems in Hilbert and in Banach spaces , 2008 .

[14]  D. Butnariu,et al.  Stable Convergence Behavior Under Summable Perturbations of a Class of Projection Methods for Convex Feasibility and Optimization Problems , 2007, IEEE Journal of Selected Topics in Signal Processing.

[15]  Yair Censor,et al.  On Diagonally Relaxed Orthogonal Projection Methods , 2007, SIAM J. Sci. Comput..

[16]  Heinz H. Bauschke,et al.  Extrapolation algorithm for affine-convex feasibility problems , 2006, Numerical Algorithms.

[17]  Dan Gordon,et al.  Component-Averaged Row Projections: A Robust, Block-Parallel Scheme for Sparse Linear Systems , 2005, SIAM J. Sci. Comput..

[18]  Gilbert Crombez,et al.  Finding common fixed points of a class of paracontractions , 2004 .

[19]  Heinz H. Bauschke,et al.  Projection and proximal point methods: convergence results and counterexamples , 2004 .

[20]  Inmaculada García,et al.  Parallel iterative reconstruction methods for structure determination of biological specimens by electron microscopy , 2003, Proceedings 2003 International Conference on Image Processing (Cat. No.03CH37429).

[21]  Yair Censor,et al.  Convergence of String-Averaging Projection Schemes for Inconsistent Convex Feasibility Problems , 2003, Optim. Methods Softw..

[22]  Ming Jiang,et al.  Convergence Studies on Iterative Algorithms for Image Reconstruction , 2003, IEEE Trans. Medical Imaging.

[23]  Heinz H. Bauschke,et al.  A Weak-to-Strong Convergence Principle for Fejé-Monotone Methods in Hilbert Spaces , 2001, Math. Oper. Res..

[24]  P. L. Combettes,et al.  Hilbertian convex feasibility problem: Convergence of projection methods , 1997 .

[25]  Patrick L. Combettes,et al.  Convex set theoretic image recovery by extrapolated iterations of parallel subgradient projections , 1997, IEEE Trans. Image Process..

[26]  Heinz H. Bauschke,et al.  On Projection Algorithms for Solving Convex Feasibility Problems , 1996, SIAM Rev..

[27]  Dimitri P. Bertsekas,et al.  On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators , 1992, Math. Program..

[28]  Yair Censor,et al.  Cyclic subgradient projections , 1982, Math. Program..

[29]  A. Lent,et al.  Iterative algorithms for large partitioned linear systems, with applications to image reconstruction , 1981 .

[30]  T. Elfving.  Block-iterative methods for consistent and inconsistent linear equations , 1980 .

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

[32]  Andrzej Cegielskiand Rafa L Zalas.  PROPERTIES OF A CLASS OF APPROXIMATELY SHRINKING OPERATORS AND THEIR APPLICATIONS , 2014 .

[33]  Touraj Nikazad,et al.  An acceleration scheme for cyclic subgradient projections method , 2013, Comput. Optim. Appl..

[34]  Andrzej Cegielski,et al.  Opial-Type Theorems and the Common Fixed Point Problem , 2011, Fixed-Point Algorithms for Inverse Problems in Science and Engineering.

[35]  L. Eisner.  Convergence of sequential and asynchronous nonlinear paracontractions , 2005 .

[36]  Gilbert Crombez,et al.  Finding common fixed points of strict paracontractions by averaging strings of sequential iterations. , 2002 .

[37]  Yair Censor,et al.  Block-Iterative Algorithms with Diagonally Scaled Oblique Projections for the Linear Feasibility Problem , 2002, SIAM J. Matrix Anal. Appl..

[38]  Y. Censor,et al.  BICAV: a block-iterative parallel algorithm for sparse systems with pixel-related weighting , 2001, IEEE Transactions on Medical Imaging.

[39]  Yair Censor,et al.  Averaging Strings of Sequential Iterations for Convex Feasibility Problems , 2001 .

[40]  P. L. Combettes,et al.  Quasi-Fejérian Analysis of Some Optimization Algorithms , 2001 .

[41]  Heinz H. Bauschke,et al.  Projection algorithms and monotone operators , 1996 .

[42]  P. L. Combettes,et al.  The Convex Feasibility Problem in Image Recovery , 1996 .

[43]  Guy Pierra,et al.  Decomposition through formalization in a product space , 1984, Math. Program..