Strong Convergence of Self-adaptive Inertial Algorithms for Solving Split Variational Inclusion Problems with Applications

In this paper, four self-adaptive iterative algorithms with inertial effects are introduced to solve a split variational inclusion problem in real Hilbert spaces. One of the advantages of the suggested algorithms is that they can work without knowing the prior information of the operator norm. Strong convergence theorems of these algorithms are established under mild and standard assumptions. As applications, the split feasibility problem and the split minimization problem in real Hilbert spaces are studied. Finally, several preliminary numerical experiments as well as an example in the field of compressed sensing are proposed to support the advantages and efficiency of the suggested methods over some existing ones.

[1]  Aviv Gibali,et al.  New self-adaptive step size algorithms for solving split variational inclusion problems and its applications , 2019, Numerical Algorithms.

[2]  Satit Saejung,et al.  Approximation of zeros of inverse strongly monotone operators in Banach spaces , 2012 .

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

[4]  Jen-Chih Yao,et al.  Inertial-Like Subgradient Extragradient Methods for Variational Inequalities and Fixed Points of Asymptotically Nonexpansive and Strictly Pseudocontractive Mappings , 2019, Mathematics.

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

[6]  B. He.  A class of projection and contraction methods for monotone variational inequalities , 1997 .

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

[8]  Suthep Suantai,et al.  Modified Proximal Algorithms for Finding Solutions of the Split Variational Inclusions , 2019, Mathematics.

[9]  L. Ceng,et al.  On general implicit hybrid iteration method for triple hierarchical variational inequalities with hierarchical variational inequality constraints , 2020 .

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

[11]  Yan Tang,et al.  Convergence analysis of a new iterative algorithm for solving split variational inclusion problems , 2020, Journal of Industrial & Management Optimization.

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

[13]  Y. Censor,et al.  Weak and Strong Convergence of Algorithms for the Split Common Null Point Problem , 2011 .

[14]  Duong Viet Thong,et al.  A new strong convergence for solving split variational inclusion problems , 2020, Numerical Algorithms.

[15]  Heinz H. Bauschke,et al.  Convex Analysis and Monotone Operator Theory in Hilbert Spaces , 2011, CMS Books in Mathematics.

[16]  Kaleem Raza Kazmi,et al.  An iterative method for split variational inclusion problem and fixed point problem for a nonexpansive mapping , 2014, Optim. Lett..

[17]  高橋 渉.  Nonlinear functional analysis : fixed point theory and its applications , 2000 .

[18]  Bing Tan,et al.  Self-adaptive inertial extragradient algorithms for solving variational inequality problems , 2020, Computational and Applied Mathematics.

[19]  L. Ceng,et al.  Hybrid inertial subgradient extragradient methods for variational inequalities and fixed point problems involving asymptotically nonexpansive mappings , 2019, Optimization.

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

[21]  L. Ceng,et al.  Generalized Mann Viscosity Implicit Rules for Solving Systems of Variational Inequalities with Constraints of Variational Inclusions and Fixed Point Problems , 2019, Mathematics.

[22]  Yekini Shehu,et al.  New inertial relaxed method for solving split feasibilities , 2020, Optimization Letters.

[23]  Marc Teboulle,et al.  A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems , 2009, SIAM J. Imaging Sci..

[24]  Pham Ky Anh,et al.  A strongly convergent Mann-type inertial algorithm for solving split variational inclusion problems , 2020 .

[25]  Abdellatif Moudafi,et al.  Split Monotone Variational Inclusions , 2011, J. Optim. Theory Appl..

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

[27]  Chih-Sheng Chuang.  Algorithms with new parameter conditions for split variational inclusion problems in Hilbert spaces with application to split feasibility problem , 2016 .

[28]  Luong Van Long,et al.  New algorithms for the split variational inclusion problems and application to split feasibility problems , 2019, Optimization.

[29]  Poom Kumam,et al.  A hybrid viscosity algorithm via modify the hybrid steepest descent method for solving the split variational inclusion in image reconstruction and fixed point problems , 2015, Appl. Math. Comput..

[30]  Giuseppe Marino,et al.  Convergence of generalized proximal point algorithms , 2004 .

[31]  Chih-Sheng Chuang.  Hybrid inertial proximal algorithm for the split variational inclusion problem in Hilbert spaces with applications , 2017 .

[32]  Hong-Kun Xu,et al.  Solving the split feasibility problem without prior knowledge of matrix norms , 2012 .

[33]  L. Ceng,et al.  Composite inertial subgradient extragradient methods for variational inequalities and fixed point problems , 2019, Journal of Inequalities and Applications.

[34]  P. Cholamjiak,et al.  Proximal type algorithms involving linesearch and inertial technique for split variational inclusion problem in hilbert spaces with applications , 2019, Optimization.

[35]  Shih-sen Chang,et al.  Strong convergence theorems for the general split variational inclusion problem in Hilbert spaces , 2013 .

[36]  C. Nahak,et al.  On inertial proximal algorithm for split variational inclusion problems , 2018, Optimization.

[37]  A. Gibali,et al.  A new relaxed CQ algorithm for solving split feasibility problems in Hilbert spaces and its applications , 2019, Journal of Industrial & Management Optimization.

[38]  F. U. Ogbuisi,et al.  An iterative method for solving split monotone variational inclusion and fixed point problems , 2016 .

[39]  Chih-Sheng Chuang.  Strong convergence theorems for the split variational inclusion problem in Hilbert spaces , 2013 .