“Optimal” choice of the step length of the projection and contraction methods for solving the split feasibility problem

In this paper, first, we review the projection and contraction methods for solving the split feasibility problem (SFP), and then by using the inverse strongly monotone property of the underlying operator of the SFP, we improve the “optimal” step length to provide the modified projection and contraction methods. Also, we consider the corresponding relaxed variants for the modified projection and contraction methods, where the two closed convex sets are both level sets of convex functions. Some convergence theorems of the proposed methods are established under suitable conditions. Finally, we give some numerical examples to illustrate that the modified projection and contraction methods have an advantage over other methods, and improve greatly the projection and contraction methods.

[1]  Deren Han,et al.  A self-adaptive projection method for solving the multiple-sets split feasibility problem , 2009 .

[2]  Yair Censor,et al.  The Subgradient Extragradient Method for Solving Variational Inequalities in Hilbert Space , 2011, J. Optim. Theory Appl..

[3]  Jigen Peng,et al.  A Cyclic and Simultaneous Iterative Method for Solving the Multiple-Sets Split Feasibility Problem , 2015, J. Optim. Theory Appl..

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

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

[6]  R. Tibshirani.  Regression Shrinkage and Selection via the Lasso , 1996 .

[7]  F. Facchinei,et al.  Finite-Dimensional Variational Inequalities and Complementarity Problems , 2003 .

[8]  Yan Gao,et al.  The strong convergence of a KM–CQ-like algorithm for a split feasibility problem , 2010 .

[9]  A. Latif,et al.  Strong convergence for generalized multiple-set split feasibility problem , 2016 .

[10]  A. Latif,et al.  A regularization algorithm for a splitting feasibility problem in Hilbert spaces , 2017 .

[11]  Q. Dong,et al.  The extragradient algorithm with inertial effects for solving the variational inequality , 2016 .

[12]  Masao Fukushima,et al.  A relaxed projection method for variational inequalities , 1986, Math. Program..

[13]  Qingzhi Yang,et al.  A simple projection method for solving the multiple-sets split feasibility problem , 2013 .

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

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

[16]  Aviv Gibali,et al.  Note on the modified relaxation CQ algorithm for the split feasibility problem , 2018, Optim. Lett..

[17]  Le Dung Muu,et al.  An algorithm for a class of split feasibility problems: application to a model in electricity production , 2016, Math. Methods Oper. Res..

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

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

[20]  Fenghui Wang,et al.  Polyak’s gradient method for split feasibility problem constrained by level sets , 2018, Numerical Algorithms.

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

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

[23]  Y. Cho,et al.  On solving split equilibrium problems and fixed point problems of nonspreading multi-valued mappings in Hilbert spaces , 2016 .

[24]  Qingzhi Yang,et al.  Self-adaptive projection methods for the multiple-sets split feasibility problem , 2011 .

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

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

[27]  Chuanxi Zhu,et al.  Iterative methods for solving the multiple-sets split feasibility problem with splitting self-adaptive step size , 2015 .

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

[29]  Yanjun Zhang,et al.  Modified projection methods for the split feasibility problem and the multiple-sets split feasibility problem , 2012, Appl. Math. Comput..

[30]  G. Papanicolaou,et al.  Enhanced statistical stability in coherent interferometric imaging , 2011 .

[31]  Hyunjoong Kim,et al.  Functional Analysis I , 2017 .

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

[33]  Y. Cho,et al.  Self-adaptive algorithms for proximal split feasibility problems and strong convergence analysis , 2015 .

[34]  Bingsheng He,et al.  On the O(1/t) convergence rate of the projection and contraction methods for variational inequalities with Lipschitz continuous monotone operators , 2013, Computational Optimization and Applications.

[35]  N. Xiu,et al.  A new halfspace-relaxation projection method for the split feasibility problem , 2008 .

[36]  Yonghong Yao,et al.  Weak convergence theorems of the modified relaxed projection algorithms for the split feasibility problem in Hilbert spaces , 2014, Optim. Lett..

[37]  Qingzhi Yang.  On variable-step relaxed projection algorithm for variational inequalities , 2005 .

[38]  Defeng Sun,et al.  A class of iterative methods for solving nonlinear projection equations , 1996 .

[39]  Muhammad Aslam Noor,et al.  On descent-projection method for solving the split feasibility problems , 2012, J. Glob. Optim..

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

[41]  Qingzhi Yang,et al.  The relaxed inexact projection methods for the split feasibility problem , 2011, Appl. Math. Comput..