Modified inertial Mann algorithm and inertial CQ-algorithm for nonexpansive mappings
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In this article, we first introduce a modified inertial Mann algorithm and an inertial CQ-algorithm by combining the accelerated Mann algorithm and the CQ-algorithm with the inertial extrapolation, respectively. This strategy is intended to speed up the convergence of the given algorithms. Then we established the convergence theorems for two provided algorithms. For the inertial CQ-algorithm, the conditions on the inertial parameters are very weak. Finally, the numerical experiments are presented to illustrate that the modified inertial Mann algorithm and inertial CQ-algorithm may have a number of advantages over other methods in computing for some cases.
KeywordsNonexpansive mapping Inertial extrapolation CQ-algorithm The inertial Mann algorithm Mann algorithm The accelerated Mann algorithm
Supported by National Natural Science Foundation of China (No. 61379102) and Open Fund of Tianjin Key Lab for Advanced Signal Processing (No. 2016ASP-TJ01). Also, the third author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and future Planning (2014R1A2A2A01002100). The authors would like to thank three anonymous referees for their careful reading of an earlier version of this paper and constructive suggestions. In particular, one referee gave us very valuable comments on the numerical experiments, which enabled us to improve the paper greatly.
- 3.Chen, P., Huang, J., Zhang, X.: A primal-dual fixed point algorithm for convex separable minimization with applications to image restoration. Inverse Probl. 29, 025011, 33 (2013)Google Scholar
- 17.Polyak, B.T: Some methods of speeding up the convergence of iteration methods, U.S.S.R. Comput. Math. Math. Phys. 4(5), 1–17 (1964)Google Scholar
- 24.Mainge, P.E: Convergence theorems for inertial KM-type algorithms. J. Comput. Appl. Math. 219, 223–236 (2008)Google Scholar
- 26.Nocedal, J., Wright, S.J.: Numerical Optimization. Springer Series in Operations Research and Financial Engineering, vol. 2, 2nd edn. Springer, Berlin (2006)Google Scholar
- 32.Dong, Q.L., Cho, Y.J., Zhong, L.L., Rassias, Th. M.: Inertial projection and contraction algorithms for variational inequalities, completedGoogle Scholar