### On Inertial Relaxation CQ Algorithm for Split Feasibility Problems

#### Abstract

In this work, we introduce an inertial relaxation CQ algorithm for the split feasibility problem in Hilbert spaces. We prove weak convergence theorem under suitable conditions. Numerical examples illustrating our methods’s efficiency are presented for comparing some known methods.

#### Keywords

#### Full Text:

PDF#### References

H. H. Bauschke and J. M. Borwein, On projection algorithms for solving convex feasibility problems, SIAM Rev. 38 (1996), 367 – 426, DOI: 10.1137/S0036144593251710.

H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, London (2011), DOI: 10.1007/978-3-319-48311-5.

H. H. Bauschke and P. L. Combettes, A weak-to-strong convergence principle for Fejér-monotone methods in Hilbert spaces, Math. Oper. Res. 26 (2001), 248 – 264, DOI: 10.1287/moor.26.2.248.10558.

R. I. Bot, E. R. Csetnek and C. Hendrich, Inertial Douglas–Rachford splitting for monotone inclusion problems, Applied Mathematics and Computation 256 (2015), 472 – 487, DOI: 10.1016/j.amc.2015.01.017.

C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Problems 18 (2002), 441 – 453, DOI: stacks.iop.org/IP/18/441.

C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems 20 (2004), 103 – 120, DOI: stacks.iop.org/IP/20/103.

Y. Censor and T. Elfving, A multiprojection algorithms using Bregman projection in a product space, Numer. Algor. 8 (1994), 221 – 239, DOI: 10.1007/BF02142692.

N. T. Vinh, P. Cholamjiak and S. Suantai, A new CQ algorithm for solving split feasibility problems in Hilbert spaces, Bulletin of the Malaysian Mathematical Sciences Society (2018), 1 – 18, DOI: 10.1007/s40840-018-0614-0.

Y. Dang, J. Sun and H. Xu, Inertial accelerated algorithms for solving a split feasibility problem, J. Ind. Manag. Optim. 13(3) (2017), 1383 – 1394, DOI: 10.3934/jimo.2016078.

J. Deepho and P. Kumam, A viscosity approximation method for the split feasibility problems, in Transactions on Engineering Technologies, G.-C. Yang, S.-I. Ao, X. Huang and O. Castillo (eds.), Springer, Dordrecht (2015), 69 – 77, DOI: 10.1007/978-94-017-9588-3_6.

Q. L. Dong, Y. C. Tang, Y. J. Cho and T. M. Rassias, “Optimal” choice of the step length of the projection and contraction methods for solving the split feasibility problem, Journal of Global Optimization 71(2) (2018), 341 – 360, DOI: 10.1007/s10898-018-0628-z.

A. Gibali, L. W. Liu and Y. C. Tang, Note on the modified relaxation CQ algorithm for the split feasibility problem, Optimization Letters 12 (2017), 1 – 14, DOI: 10.1007/s11590-017-1148-3.

W. Kumam, J. Deepho and P. Kumam, Hybrid extragradient method for finding a common solution of the split feasibility and system of equilibrium problems, Dyn. Contin. Discrete Impuls. Syst., Ser. B, Appl. Algorithms 21(6) (2014), 367 – 388.

P. E. Mainge, Convergence theorem for inertial KM-type algorithms, Journal of Computational and Applied Mathematics 219 (2008), 223 – 236, DOI: 10.1016/j.cam.2007.07.021.

B. Qu and N. Xiu, A note on the CQ algorithm for the split feasibility problem, Inverse Problems 21 (2005), 1655 – 1665, DOI: 10.1088/0266-5611/21/5/009.

H. Stark, Image Recovery: Theory and Application, Elsevier (2013).

K. Sitthithakerngkiet, J. Deepho and P. Kumam, Modified hybrid steepest method for the split feasibility problem in image recovery of inverse problems, Numerical Functional Analysis and Optimization 38(4) (2017), 507 – 522, DOI: 10.1080/01630563.2017.1287084.

S. Suantai, Y. Shehu, P. Cholamjiak and O. S. Iyiola, Strong convergence of a self-adaptive method for the split feasibility problem in Banach spaces, Journal of Fixed Point Theory and Applications 20(2) (2018), 68, DOI: 10.1007/s11784-018-0549-y.

S. Suantai, Y. Shehu and P. Cholamjiak, Nonlinear iterative methods for solving the split common null point problem in Banach spaces, Optimization Methods and Software 34(4) (2019), 853 – 874, DOI: 10.1080/10556788.2018.1472257.

S. Suantai, N. Pholasa and P. Cholamjiak, The modified inertial relaxed CQ algorithm for solving the split feasibility problems, Journal of Industrial and Management Optimization 14(4) (2018), 1595 – 1615, DOI: 10.3934/jimo.2018023.

R. Tibshirani, Regression shrinkage and selection via the lasso, J. R. Stat. Soc. Ser. B. Stat. Methodol. 58 (1996), 267 – 288, DOI: 10.1111/j.2517-6161.1996.tb02080.x.

Q. Yang, The relaxed CQ algorithm for solving the split feasibility problem, Inverse Problems 20 (2004), 1261 – 1266, DOI: 10.1088/0266-5611/20/4/014.

W. Zhang, D. Han and Z. Li, A self-adaptive projection method for solving the multiple-sets split feasibility problem, Inverse Problems 25 (2009), 115001, DOI: 10.1088/0266-5611/27/3/035009.

### Refbacks

- There are currently no refbacks.

eISSN 0975-8607; pISSN 0976-5905