Solving a Split Feasibility Problem by the Strong Convergence of Two Projection Algorithms in Hilbert Spaces

The goal of this manuscript is to establish strong convergence theorems for inertial shrinking projection and CQ algorithms to solve a split convex feasibility problem in real Hilbert spaces. Finally, numerical examples were obtained to discuss the performance and effectiveness of our algorithms and compare the proposed algorithms with the previous shrinking projection, hybrid projection, and inertial forward-backward methods.

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

[2]  Y. Shehu,et al.  A strong convergence result involving an inertial forward–backward algorithm for monotone inclusions , 2017 .

[3]  A. Moudafi,et al.  Convergence of a splitting inertial proximal method for monotone operators , 2003 .

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

[5]  H. H. Rachford,et al.  On the numerical solution of heat conduction problems in two and three space variables , 1956 .

[6]  Hong-Kun Xu,et al.  Averaged Mappings and the Gradient-Projection Algorithm , 2011, J. Optim. Theory Appl..

[7]  Boris Polyak.  Some methods of speeding up the convergence of iteration methods , 1964 .

[8]  Manuel De la Sen,et al.  Shrinking Projection Methods for Accelerating Relaxed Inertial Tseng-Type Algorithm with Applications , 2020 .

[9]  H. Attouch,et al.  An Inertial Proximal Method for Maximal Monotone Operators via Discretization of a Nonlinear Oscillator with Damping , 2001 .

[10]  Ana Alonso Rodríguez,et al.  A discontinuous Galerkin method for a time-harmonic eddy current problem , 2017, Calcolo.

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

[12]  Gregory B. Passty.  Ergodic convergence to a zero of the sum of monotone operators in Hilbert space , 1979 .

[13]  Xiaolong Qin,et al.  Convergence of splitting algorithms for the sum of two accretive operators with applications , 2014 .

[14]  Hong-Kun Xu,et al.  Strong convergence of the CQ method for fixed point iteration processes , 2006 .

[15]  Paul Tseng,et al.  A Modified Forward-backward Splitting Method for Maximal Monotone Mappings 1 , 1998 .

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

[17]  P. Lions,et al.  Splitting Algorithms for the Sum of Two Nonlinear Operators , 1979 .

[18]  Kong,et al.  A NOTE ON THE GRADIENT PROJECTION METHOD WITH EXACT STEPSIZE RULE *1) , 2007 .

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

[20]  H. H. Rachford,et al.  The Numerical Solution of Parabolic and Elliptic Differential Equations , 1955 .

[21]  Manuel de la Sen,et al.  Advanced Algorithms and Common Solutions to Variational Inequalities , 2020, Symmetry.

[22]  W. Takahashi,et al.  Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups , 2003 .

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

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

[25]  Wu Jigang,et al.  A Regularized Gradient Projection Method for the Minimization Problem , 2012, J. Appl. Math..

[26]  Y. J. Cho,et al.  Modified inertial Mann algorithm and inertial CQ-algorithm for nonexpansive mappings , 2018, Optim. Lett..

[27]  Y. Cho,et al.  General Inertial Mann Algorithms and Their Convergence Analysis for Nonexpansive Mappings , 2018 .

[28]  Hong-Kun Xu,et al.  Forward-Backward Splitting Methods for Accretive Operators in Banach Spaces , 2012 .

[29]  Duong Viet Thong,et al.  Tseng type methods for solving inclusion problems and its applications , 2018, Calcolo.

[30]  Q. Dong,et al.  A new hybrid algorithm for a nonexpansive mapping , 2015 .

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

[32]  Huanhuan Cui,et al.  On the contraction-proximal point algorithms with multi-parameters , 2012, J. Glob. Optim..

[33]  Xiaoqun Zhang,et al.  A primal–dual fixed point algorithm for convex separable minimization with applications to image restoration , 2013 .

[34]  Y. Shehu.  Iterative Approximations for Zeros of Sum of Accretive Operators in Banach Spaces , 2016 .

[35]  Y. Nesterov.  A method for solving the convex programming problem with convergence rate O(1/k^2) , 1983 .

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

[37]  Jie Sun,et al.  Inertial accelerated algorithms for solving a split feasibility problem , 2016 .