STRONG CONVERGENCE OF A GENERAL VISCOSITY EXPLICIT RULE FOR THE SUM OF TWO MONOTONE OPERATORS IN HILBERT SPACES

Prasit Cholamjiak; Suthep Suantai; Pongsakorn Sunthrayuth

doi:10.11948/20180191

2019 Volume 9 Issue 6

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Prasit Cholamjiak, Suthep Suantai, Pongsakorn Sunthrayuth. STRONG CONVERGENCE OF A GENERAL VISCOSITY EXPLICIT RULE FOR THE SUM OF TWO MONOTONE OPERATORS IN HILBERT SPACES[J]. Journal of Applied Analysis & Computation, 2019, 9(6): 2137-2155. doi: 10.11948/20180191

Citation:

Prasit Cholamjiak, Suthep Suantai, Pongsakorn Sunthrayuth. STRONG CONVERGENCE OF A GENERAL VISCOSITY EXPLICIT RULE FOR THE SUM OF TWO MONOTONE OPERATORS IN HILBERT SPACES[J]. Journal of Applied Analysis & Computation, 2019, 9(6): 2137-2155. doi: 10.11948/20180191

STRONG CONVERGENCE OF A GENERAL VISCOSITY EXPLICIT RULE FOR THE SUM OF TWO MONOTONE OPERATORS IN HILBERT SPACES

1.
School of Science, University of Phayao, Phayao 56000, Thailand
2.
Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
3.
Department of Mathematics and Computer Science, Faculty of Science and Technology, Rajamangala University of Technology Thanyaburi (RMUTT), Pathumthani, Thailand

Corresponding author: Email address: pongsakorn su@rmutt.ac.th (P. Sunthrayuth)

Abstract

Abstract

In this paper, we study a general viscosity explicit rule for approximating the solutions of the variational inclusion problem for the sum of two monotone operators. We then prove its strong convergence under some new conditions on the parameters in the framework of Hilbert spaces. As applications, we apply our main result to the split feasibility problem and the LASSO problem. We also give some numerical examples to support our main result. The results presented in this paper extend and improve the corresponding results in the literature.
- Monotone operator /
- Hilbert space /
- strong convergence /
- iterative method
MSC: 47H09, 47H10, 47J25, 47J05

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References

[1]	N. Abdou, B.A. Alamri, Y.J. Cho, Y. Yao and L.J. Zhu, Parallel algorithms for variational inclusions and fixed points with applications, Fixed Point Theory Appl., 2014, 2014(174). Google Scholar
[2]	A. Alghamdi, N. Shahzad and H.K. Xu, Properties and iterative methods for the $Q$-Lasso, Abst. Appl. Anal., 2013, Art ID 250943, 8 pages. Google Scholar
[3]	B. Bassty, Ergodic convergence to a zero of the sum of monotone operators in Hilbert space, J. Math. Anal. Appl., 1979, 72, 383-390. doi: 10.1016/0022-247X(79)90234-8 CrossRef Google Scholar
[4]	P. Bertsekas and J.N. Tsitsiklis, Parallel and Distributed Computation, Numerical Methods. Athena Scientific. Belmont. MA, 1997. Google Scholar
[5]	H. Brézis and P.L. Lions, Produits infinis de resolvantes, Israel J. Math., 1978, 29, 329-345. doi: 10.1007/BF02761171 CrossRef Google Scholar
[6]	C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Probl., 2004, 20, 103-120. doi: 10.1088/0266-5611/20/1/006 CrossRef Google Scholar
[7]	C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Probl., 2002, 18, 441-453. Google Scholar
[8]	C. Ceng, Approximation of common solutions of a split inclusion problem and a fixed-point problem, J. Appl. Numer. Optim., 2019, 1, 1-12. Google Scholar
[9]	Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algor., 1994, 8, 221-239. doi: 10.1007/BF02142692 CrossRef Google Scholar
[10]	Y. Censor, T. Elfving, N. Kopf and T. Bortfeld, The multiple-sets split feasibility problem and its applications for inverse problems, Inverse Probl., 2005, 21, 2071-2084. doi: 10.1088/0266-5611/21/6/017 CrossRef Google Scholar
[11]	Y. Censor and A. Segal, The split common fixed point problem for directed operators. J. Convex Anal., 2009, 16, 587-600. Google Scholar
[12]	S. Chang, C.-F. Wen and J.-C. Yao, Zero point problem of accretive operators in Banach spaces, Bull. Malays. Math. Sci. Soc., 2019, 42, 105-118. doi: 10.1007/s40840-017-0470-3 CrossRef Google Scholar
[13]	G. Chen and R.T. Rockafellar, Convergence rates in forward-backward splitting, SIAM J. Optim., 1997, 7, 421-444. doi: 10.1137/S1052623495290179 CrossRef Google Scholar
[14]	Y. Cho, X. Qin and L. Wang, Strong convergence of a splitting algorithm for treating monotone operators, Fixed Point Theory Appl., 2014, Art ID 94. Google Scholar
[15]	P. Cholamjiak and Y. Shehu, Inertial forward-backward splitting method in Banach spaces with application to compressed sensing, Appl. Math., 2019, 64, 409-435. doi: 10.21136/AM.2019.0323-18 CrossRef Google Scholar
[16]	L. Combettes and V.R. Wajs, Signal recovery by proximal forward-backward splitting, Multiscale Model. Simul., 2005, 4, 1168-1200. doi: 10.1137/050626090 CrossRef Google Scholar
[17]	C. Dunn, Convexity, monotonicity, and gradient processes in Hilbert space, J. Math. Anal. Appl., 1976, 53, 145-158. Google Scholar
[18]	O. Güler, On the convergence of the proximal point algorithm for convex minimization, SIAM J. Control Optim., 1991, 29, 403-419. doi: 10.1137/0329022 CrossRef Google Scholar
[19]	S. Kamimura and W. Takahashi, Approximating solutions of maximal monotone operators in Hilbert spaces, J. Approx. Theory, 2000, 106, 226-240. doi: 10.1006/jath.2000.3493 CrossRef Google Scholar
[20]	K. Kankam, N. Pholasa and P. Cholamjiak, On convergence and complexity of the modified forward-backward method involving new linesearches for convex minimization, Math. Meth. Appl. Sci., 2019, 42, 1352-1362. doi: 10.1002/mma.5420 CrossRef Google Scholar
[21]	L. Lions and B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM J. Numer. Anal., 1979, 16, 964-979. doi: 10.1137/0716071 CrossRef Google Scholar
[22]	J. Lin and W. Takahashi, A general iterative method for hierarchical variational inequality problems in Hilbert spaces and applications, Positivity, 2012, 16, 429-453. doi: 10.1007/s11117-012-0161-0 CrossRef Google Scholar
[23]	G. López, V. Martín-Márquez, F. Wang and H.-K. Xu, Forward-backward splitting methods for accretive operators in Banach spaces, Abstr. Appl. Anal., 2012, Art ID 109236. Google Scholar
[24]	E. Maingé, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal., 2008, 16, 899-912. doi: 10.1007/s11228-008-0102-z CrossRef Google Scholar
[25]	G. Marino, A. Rugiano and D.R. Sahu, Strong convergence for a general explicit convex combination method for nonexpansive mappings and equilibrium points, J. Nonlinear Convex Anal., 2017, 18, 1953-1966. Google Scholar
[26]	G. Marino and H.-K. Xu, A general iterative method for nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 2006, 318, 43-52. doi: 10.1016/j.jmaa.2005.05.028 CrossRef Google Scholar
[27]	B. Martinet, Régularisation d'inéquations variationnelles par approximations successives, Rev. Française Informat. Recherche. Opérationnelle., 1970, 4, 154-158. Google Scholar
[28]	E. Masad and S. Reich, A note on the multiple-set split convex feasibility problem in Hilbert space. J. Nonlinear Convex Anal., 2007, 8, 367-371. Google Scholar
[29]	A. Moudafi, Viscosity approximation methods for fixed point problems, J. Math. Anal. Appl., 2000, 241, 46-55. doi: 10.1006/jmaa.1999.6615 CrossRef Google Scholar
[30]	N. Pholasa and P. Cholamjiak, The Regularization Method for Solving Variational Inclusion Problems, Thai J. Math., 2016, 14, 369-381. Google Scholar
[31]	X. Qin, S.Y. Cho and L. Wang, A regularization method for treating zero points of the sum of two monotone operators, Fixed Point Theory Appl., 2014, Art ID 75. Google Scholar
[32]	T. Rockafellar, Monotone operators and the proximal point algorithm. SIAM J. Control Optim., 1976, 14, 877-898. Google Scholar
[33]	T. Rockafellar, On the maximality of subdifferential mappings, Pac. J. Math., 1970, 33, 209-216. doi: 10.2140/pjm.1970.33.209 CrossRef Google Scholar
[34]	S. Sra, S. Nowozin and S.J. Wright, Optimization for machine learning, Cambridge, MIT Press, 2012. Google Scholar
[35]	P. Sunthrayuth and P. Kumam, The resolvent operator techniques with perturbations for finding zeros of maximal monotone operator and fixed point problems in Hilbert spaces, Thai J. Math., 2016, 14, 1-21. Google Scholar
[36]	W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, 2000. Google Scholar
[37]	S. Takahashi, W. Takahashi and M. Toyoda, Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces, J. Optim. Theory Appl., 2010, 147, 27-41. doi: 10.1007/s10957-010-9713-2 CrossRef Google Scholar
[38]	V. Thong and P. Cholamjiak, Strong convergence of a forward-backward splitting method with a new step size for solving monotone inclusions, Comput. Appl. Math., 2019, 38. DOI: 10.1007/s40314-019-0855-z. Google Scholar
[39]	R. Tibshirani, Regression shrinkage and selection via the lasso. J Roy. Stat. Soc. Ser. B., 1996, 58, 267-288. Google Scholar
[40]	P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control Optim., 2000, 38, 431-446. doi: 10.1137/S0363012998338806 CrossRef Google Scholar
[41]	F. Wang and H.-K. Xu, Cyclic algorithms for split feasibility problems in Hilbert spaces, Nonlinear Anal., 2011, 74, 4105-4111. doi: 10.1016/j.na.2011.03.044 CrossRef Google Scholar
[42]	K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 2002, 66, 240-256. doi: 10.1112/S0024610702003332 CrossRef Google Scholar
[43]	K. Xu, Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces, Inverse Probl., 2010, 26, 105018, 17 pages. Google Scholar
[44]	K. Xu, Properties and iterative methods for the Lasso and its variants, Chin. Ann. Math., 2014, 35B(3), 501-518. Google Scholar
[45]	Q. Yang, The relaxed CQ algorithm solving the split feasibility problem, Inverse Probl., 2004, 20, 1261-1266. doi: 10.1088/0266-5611/20/4/014 CrossRef Google Scholar
[46]	J. Zhao and Q. Yang, Several solution methods for the split feasibility problem, Inverse Probl., 2005, 21, 1791-1799. doi: 10.1088/0266-5611/21/5/017 CrossRef Google Scholar