# Relaxed inertial accelerated algorithms for solving split equality feasibility problem

Volume 10, Issue 8, pp 4109--4121 Publication Date: August 07, 2017

### Authors

Meixia Li - School of Mathematics and Information Science, Weifang University, Weifang, 261061, China.
Xiping Kao - College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, China.
Haitao Che - School of Mathematics and Information Science, Weifang University, Weifang, 261061, China.

### Abstract

In this paper, we study the split equality feasibility problem and present two algorithms for solving the problem with special structure. We prove the weak convergence of these algorithms under mild conditions. Especially, the selection of stepsize is only dependent on the information of current iterative points, but independent from the prior knowledge of operator norms. These algorithms provide new ideas for solving the split equality feasibility problem. Numerical results demonstrate the feasibility and effectiveness of these algorithms.

### Keywords

• Split equality feasibility problem
• relaxed inertial accelerated algorithm
• weak convergence
• subdifferential.

### References

[1] H. H. Bauschke, J. M. Borwein, On projection algorithms for solving convex feasibility problems, SIAM Rev., 38 (1996), 367–426.
[2] C. L. Byrne, A. Moudafi, Extensions of the CQ Algorithm for the split feasibility and split equality problems (10th draft), hal-00776640-version 1, (2012).
[3] Y. Censor, Parallel application of block-iterative methods in medical imaging and radiation therapy, Math. Programming, 42 (1998), 307–325.
[4] Y. Censor, T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms, 8 (1994), 221–239.
[5] S.-S. Chang, Some problems and results in the study of nonlinear analysis, Proceedings of the Second World Congress of Nonlinear Analysts, Part 7, Athens, (1996), Nonlinear Anal., 30 (1997), 4197–4208.
[6] S.-S. Chang, L.Wang, L.-J. Qin, Split equality fixed point problem for quasi-pseudo-contractive mappings with applications, Fixed Point Theory Appl., 2015 (2015), 12 pages.
[7] H.-T. Che, M.-X. Li, A simultaneous iterative method for split equality problems of two finite families of strictly pseudononspreading mappings without prior knowledge of operator norms, Fixed Point Theory Appl., 2015 (2015), 14 pages.
[8] F. H. Clarke, Optimization and nonsmooth analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, (1983).
[9] Y.-Z. Dang, J. Sun, H.-L. Xu, Inertial accelerated algorithms for solving a split feasibility problem, J. Ind. Manag. Optim., 13 (2017), 1383–1394.
[10] F. Deutsch, The method of alternating orthogonal projections, Approximation theory, spline functions and applications, Maratea, (1991), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Kluwer Acad. Publ., Dordrecht, 356 (1992), 105–121.
[11] M. Fukushima, A relaxed projection method for variational inequalities, Math. Programming, 35 (1986), 58–70.
[12] Y. Gao, Piecewise smooth Lyapunov function for a nonlinear dynamical system, J. Convex Anal., 19 (2012), 1009–1015.
[13] G. T. Herman, Image reconstruction from projections: the fundamentals of computerized tomography, Academic Press, New York, (1980).
[14] P. E. Maing´e, Convergence theorems for inertial KM-type algorithms, J. Comput. Appl. Math., 219 (2008), 223–236.
[15] A. Moudafi, Alternating CQ-algorithms for convex feasibility and split fixed-point problems, J. Nonlinear Convex Anal., 15 (2014), 809–818.
[16] B. T. Polyak, Minimization of unsmooth functionals, USSR Comput. Math. Math. Phys., 9 (1969), 14–29.
[17] B. Qu, N.-H. Xiu, A new halfspace-relaxation projection method for the split feasibility problem, Linear Algebra Appl., 428 (2008), 1218–1229.
[18] R. T. Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28 Princeton University Press, Princeton, N.J., (1970).
[19] H.-K. Xu, A variable Krasnoselski˘ı-Mann algorithm and the multiple-set split feasibility problem, Inverse Problems, 22 (2006), 2021–2034.
[20] A.-L. Yan, G.-Y. Wang, N.-H. Xiu, Robust solutions of split feasibility problem with uncertain linear operator, J. Ind. Manag. Optim., 3 (2007), 749–761.
[21] Q.-Z. Yang, The relaxed CQ algorithm solving the split feasibility problem, Inverse Problems, 20 (2004), 1261–1266.
[22] J.-L. Zhao, Q.-Z. Yang, Several solution methods for the split feasibility problem, Inverse Problems, 21 (2005), 1791–1799.