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## A novel algorithm for approximating common solution of a system of monotone inclusion problems and common fixed point problem

 1 Department of Mathematics, University of Science and Technology of Mazandaran, Behshahr, Iran 2 School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran

Received  July 2021 Revised  October 2021 Early access December 2021

Fund Project: This research was in part supported by a grant from IPM (No.1400470032)

In this paper, we study the problem of finding a common element of the set of solutions of a system of monotone inclusion problems and the set of common fixed points of a finite family of generalized demimetric mappings in Hilbert spaces. We propose a new and efficient algorithm for solving this problem. Our method relies on the inertial algorithm, Tseng's splitting algorithm and the viscosity algorithm. Strong convergence analysis of the proposed method is established under standard and mild conditions. As applications we use our algorithm for finding the common solutions to variational inequality problems, the constrained multiple-set split convex feasibility problem, the convex minimization problem and the common minimizer problem. Finally, we give some numerical results to show that our proposed algorithm is efficient and implementable from the numerical point of view.

Citation: Mohammad Eslamian, Ahmad Kamandi. A novel algorithm for approximating common solution of a system of monotone inclusion problems and common fixed point problem. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021210
##### References:
 [1] T. O. Alakoya, A. O. E. Owolabi and O. T. Mewomo, An inertial algorithm with a selfadaptive step size for a split equilibrium problem and a fixed point problem of an infinite family of strict pseudo-contractions, J. Nonlinear Var. Anal., 5 (2021), 803-829. [2] T. O. Alakoya, A. Taiwo, O. T. Mewomo and Y. J. Cho, An iterative algorithm for solving variational inequality, generalized mixed equilibrium, convex minimization and zeros problems for a class of nonexpansive-type mappings, Ann. Univ. Ferrara Sez. Ⅶ Sci. Mat., 67 (2021), 1-31.  doi: 10.1007/s11565-020-00354-2. [3] C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities. Applications to Free Boundary Problems, Wiley, New York, 1984. [4] H. Brzis and I. I. Chapitre, Operateurs maximaux monotones, North-Holland Math. Stud., 5 (1973), 19-51. [5] C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Problems, 18 (2002), 441-453.  doi: 10.1088/0266-5611/18/2/310. [6] C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20 (2004), 103-120.  doi: 10.1088/0266-5611/20/1/006. [7] A. Cegielski, Iterative Methods for Fixed Point Problems in Hilbert Spaces, Lecture Notes in Mathematics, 2057, Springer, Heidelberg, 2012. [8] L. C. Ceng, A. Petrusel, X. Qin and J. C. Yao, A modified inertial subgradient extragradient method for solving pseudomonotone variational inequalities and common fixed point problems, Fixed Point Theory, 21 (2020), 93-108.  doi: 10.24193/fpt-ro.2020.1.07. [9] L. C. Ceng, A. Petrusel, X. Qin and J. C. Yao, Two inertial subgradient extragradient algorithms for variational inequalities with fixed-point constraints, Optimization, 70 (2021), 1337-1358.  doi: 10.1080/02331934.2020.1858832. [10] L. C. Ceng, A. Petrusel, J. C. Yao and Y. Yao, Hybrid viscosity extragradient method for systems of variational inequalities, fixed points of nonexpansive mappings, zero points of accretive operators in Banach spaces, Fixed Point Theory, 19 (2018), 487-501.  doi: 10.24193/fpt-ro.2018.2.39. [11] L. C. Ceng and M. Shang, Composite extragradient implicit rule for solving a hierarch variational inequality with constraints of variational inclusion and fixed point problems, J. Inequal. Appl., (2020), 33, 19pp. doi: 10.1186/s13660-020-2306-1. [12] L. C. Ceng and M. Shang, Hybrid inertial subgradient extragradient methods for variational inequalities and fixed point problems involving asymptotically nonexpansive mappings, Optimization, 70 (2021), 715-740.  doi: 10.1080/02331934.2019.1647203. [13] Y. Censor, T. Bortfeld, B. Martin and A. Trofimov, A unified approach for inversion problems in intensity-modulated radiation therapy, Phys. Med. Biol., 51 (2006), 2353-2365.  doi: 10.1088/0031-9155/51/10/001. [14] Y. Censor and T. Elfving, A multiprojection algorithms using Bragman projection in a product space, Numer. Algorithms, 8 (1994), 221-239.  doi: 10.1007/BF02142692. [15] Y. Censor, T. Elfving, N. Kopf and T. Bortfeld, The multiple-sets split feasibility problem and its applications, Inverse Problems, 21 (2005), 2071-2084.  doi: 10.1088/0266-5611/21/6/017. [16] Y. Censor, A. Gibali, S. Reich and S. Sabach, Common solutions to variational inequalities, Set-Valued Var. Anal., 20 (2012), 229-247.  doi: 10.1007/s11228-011-0192-x. [17] H. G. Chen and R. T. Rockafellar, Convergence rates in forward-backward splitting, SIAM J. Optim., 7 (1997), 421-444.  doi: 10.1137/S1052623495290179. [18] S. Chen, D. Donoho and M. Saunders, Atomic decomposition by basis pursuit, SIAM J. Sci. Comput., 20 (1998), 33-61.  doi: 10.1137/S1064827596304010. [19] P. L. Combettes and V. R. Wajs, Signal recovery by proximal forward-backward splitting, Multiscale Model. Simul., 4 (2005), 1168-1200.  doi: 10.1137/050626090. [20] I. Daubechies, M. Defrise and C. De Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Commun. Pure Appl. Math., 57 (2004), 1413-1457.  doi: 10.1002/cpa.20042. [21] Q. L. Dong, Y. J. Cho, L. L. Zhong and T. M. Rassias, Inertial projection and contraction algorithms for variational inequalities, J. Global Optim., 70 (2018), 687-704.  doi: 10.1007/s10898-017-0506-0. [22] J. Duchi and Y. Singer, Efficient online and batch learning using forward-backward splitting, J. Mach. Learn. Res., 10 (2009), 2899-2934. [23] M. Eslamian, Strong convergence theorem for common zero points of inverse strongly monotone mappings and common fixed points of generalized demimetric mappings, Optimization, 2021. doi: 10.1080/02331934.2021.1939341. [24] M. Eslamian, Y. Shehu and O. S. Iyiola, A novel iterative algorithm with convergence analysis for split common fixed points and variational inequality prblems, Fixed Point Theory, 22 (2021), 123-140.  doi: 10.24193/fpt-ro.2021.1.09. [25] A. Gibali and D. V. Thong, Tseng type methods for solving inclusion problems and its applications, Calcolo, 55 (2018), Article ID 49. doi: 10.1007/s10092-018-0292-1. [26] R. Glowinski, J. L. Lions and R. Tremolieres, Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, 1981. [27] S. He and C. Yang, Solving the variational inequality problem defined on intersectoin of finite level sets, Abstr. Appl. Anal., 2013 (2013), Article ID 942315, 8pp. doi: 10.1155/2013/942315. [28] T. Kawasaki and W. Takahashi, A strong convergence theorem for countable families of nonlinear nonself mappings in Hilbert spaces and applications, J. Nonlinear Convex Anal., 19 (2018), 543-560. [29] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980. [30] D. Kitkuan, P. Kumam, J. Martínez-Moreno and K. Sitthithakerngkiet, Inertial viscosity forward-backward splitting algorithm for monotone inclusions and its application to image restoration problems, Int. J. Comput. Math., 97 (2019), 482-497.  doi: 10.1080/00207160.2019.1649661. [31] P. L. Lions and B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM J. Numer. Anal., 16 (1979), 964-979.  doi: 10.1137/0716071. [32] D. A. Lorenz and T. Pock, An inertial forward-backward algorithm for monotone inclusions, J. Math. Imaging Vision, 51 (2015), 311-325.  doi: 10.1007/s10851-014-0523-2. [33] E. Masad and S. Reich, A note on the multiple-set split convex feasibility problem, J. Nonlinear Convex Anal., 8 (2007), 367-371. [34] A. Moudafi, Vicosity approximation methods for fixed point problems, J. Math. Anal. Appl., 241 (2000), 46-55.  doi: 10.1006/jmaa.1999.6615. [35] Y. Nesterov, A method for solving the convex programming problem with convergence rate O(1/k2), Dokl. Akad. Nauk SSSR, 269 (1983), 543-547. [36] G. N. Ogwo, T. O. Alakoya and O. T. Mewomo, Iterative algorithm with self-adaptive step size for approximating the common solution of variational inequality and fixed point problems, Optimization, 2021. doi: 10.1080/02331934.2021.1981897. [37] G. N. Ogwo, C. Izuchukwu and O. T. Mewomo, Inertial methods for finding minimumnorm solutions of the split variational inequality problem beyond monotonicity, Numer. Algorithms, 88 (2021), 1419-1456.  doi: 10.1007/s11075-021-01081-1. [38] M. A. Olona, T. O. Alakoya, A. O.-E. Owolabi and O. T. Mewomo, Inertial shrinking projection algorithm with self-adaptive step size for split generalized equilibrium and fixed point problems for a countable family of nonexpansive multivalued mappings, Demonstr. Math., 54 (2021), 47-67.  doi: 10.1515/dema-2021-0006. [39] A. O. E. Owolabi, T. O. Alakoya, A. Taiwo and O. T. Mewomo, A new inertial-projection algorithm for approximating common solution of variational inequality and fixed point problems of multivalued mappings, Numer. Algebra Control Optim., 2021 doi: 10.3934/naco.2021004. [40] N. Parikh and S. Boyd, Proximal Algorithms, Found. Trends Optim., 2014. doi: 10.1561/9781601987174. [41] G. B. Passty, Ergodic convergence to a zero of the sum of monotone operators in Hilbert spaces, J. Math. Anal. Appl., 72 (1979), 383-390.  doi: 10.1016/0022-247X(79)90234-8. [42] B. T. Polyak, Some methods of speeding up the convergence of iteration methods, Ž. Vyčisl. Mat i Mat. Fiz., 4 (1964), 791-803. [43] H. Raguet, J. Fadili and G. Peyré, A generalized forward-backward splitting, SIAM J. Imaging Sci., 6 (2013), 1199-1226.  doi: 10.1137/120872802. [44] R. T. Rockafellar, Characterization of the subdifferentials of convex functions, Pacific J. Math., 17 (1966), 497-510.  doi: 10.2140/pjm.1966.17.497. [45] R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), 877-898.  doi: 10.1137/0314056. [46] W. Takahashi, Nonlinear Functional Analysis-Fixed Point Theory and Its Applications, Yokohama Publishers, Yokohama, 2000. [47] W. Takahashi, Strong convergence theorem for a finite family of demimetric mappings with variational inequality problems in a Hilbert space, Jpn. J. Ind. Appl. Math., 34 (2017), 41-57.  doi: 10.1007/s13160-017-0237-0. [48] W. Takahashi, The split common fixed point problem and the shrinking projection method in Banach spaces, J. Convex Anal., 24 (2017), 1015-1028. [49] W. Takahashi, Weak and strong convergence theorems for new demimetric mappings and the split common fixed point problem in Banach spaces, Numer. Funct. Anal. Appl. Optim., 39 (2018), 1011-1033.  doi: 10.1080/01630563.2018.1466803. [50] B. Tan and S. Y. Cho, Strong convergence of inertial forward backward methods for solving monotone inclusions, Applicable Analysis, 2021. doi: 10.1080/00036811.2021.1892080. [51] B. Tan, X. Qin and J. C. Yao, Strong convergence of self-adaptive inertial algorithms for solving split variational inclusion problems with applications, J. Sci. Comput., 87 (2021), Article ID 20, 34pp. doi: 10.1007/s10915-021-01428-9. [52] M. Tian and G. Xu, Inertial modified projection algorithm with self-adaptive technique for solving pseudo-monotone variational inequality problems in Hilbert spaces, Optimization, 2021. doi: 10.1080/02331934.2021.1928123. [53] R. Tibshirami, Regression shrinkage and selection via lasso, J. Roy. Statist. Soc. Ser. B, 58 (1996), 267-288.  doi: 10.1111/j.2517-6161.1996.tb02080.x. [54] P. Tseng, A modified forward backward splitting method for maximal monotone mappings, SIAM J. Control Optim., 38 (2000), 431-446.  doi: 10.1137/S0363012998338806. [55] E. Zeidler, Nonlinear Functional Analysis and Its Applications Ⅲ: Variational Methods and Optimization, Springer-Verlag, New York, 1985. [56] T. Y. Zhao, D. Q. Wang and L. C. Ceng et al., Inertial method for split null point problems with pseudomonotone variational inequality problems, Numer. Funct. Anal. Appl., 42 (2020), 69-90.

show all references

##### References:
 [1] T. O. Alakoya, A. O. E. Owolabi and O. T. Mewomo, An inertial algorithm with a selfadaptive step size for a split equilibrium problem and a fixed point problem of an infinite family of strict pseudo-contractions, J. Nonlinear Var. Anal., 5 (2021), 803-829. [2] T. O. Alakoya, A. Taiwo, O. T. Mewomo and Y. J. Cho, An iterative algorithm for solving variational inequality, generalized mixed equilibrium, convex minimization and zeros problems for a class of nonexpansive-type mappings, Ann. Univ. Ferrara Sez. Ⅶ Sci. Mat., 67 (2021), 1-31.  doi: 10.1007/s11565-020-00354-2. [3] C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities. Applications to Free Boundary Problems, Wiley, New York, 1984. [4] H. Brzis and I. I. Chapitre, Operateurs maximaux monotones, North-Holland Math. Stud., 5 (1973), 19-51. [5] C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Problems, 18 (2002), 441-453.  doi: 10.1088/0266-5611/18/2/310. [6] C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20 (2004), 103-120.  doi: 10.1088/0266-5611/20/1/006. [7] A. Cegielski, Iterative Methods for Fixed Point Problems in Hilbert Spaces, Lecture Notes in Mathematics, 2057, Springer, Heidelberg, 2012. [8] L. C. Ceng, A. Petrusel, X. Qin and J. C. Yao, A modified inertial subgradient extragradient method for solving pseudomonotone variational inequalities and common fixed point problems, Fixed Point Theory, 21 (2020), 93-108.  doi: 10.24193/fpt-ro.2020.1.07. [9] L. C. Ceng, A. Petrusel, X. Qin and J. C. Yao, Two inertial subgradient extragradient algorithms for variational inequalities with fixed-point constraints, Optimization, 70 (2021), 1337-1358.  doi: 10.1080/02331934.2020.1858832. [10] L. C. Ceng, A. Petrusel, J. C. Yao and Y. Yao, Hybrid viscosity extragradient method for systems of variational inequalities, fixed points of nonexpansive mappings, zero points of accretive operators in Banach spaces, Fixed Point Theory, 19 (2018), 487-501.  doi: 10.24193/fpt-ro.2018.2.39. [11] L. C. Ceng and M. Shang, Composite extragradient implicit rule for solving a hierarch variational inequality with constraints of variational inclusion and fixed point problems, J. Inequal. Appl., (2020), 33, 19pp. doi: 10.1186/s13660-020-2306-1. [12] L. C. Ceng and M. Shang, Hybrid inertial subgradient extragradient methods for variational inequalities and fixed point problems involving asymptotically nonexpansive mappings, Optimization, 70 (2021), 715-740.  doi: 10.1080/02331934.2019.1647203. [13] Y. Censor, T. Bortfeld, B. Martin and A. Trofimov, A unified approach for inversion problems in intensity-modulated radiation therapy, Phys. Med. Biol., 51 (2006), 2353-2365.  doi: 10.1088/0031-9155/51/10/001. [14] Y. Censor and T. Elfving, A multiprojection algorithms using Bragman projection in a product space, Numer. Algorithms, 8 (1994), 221-239.  doi: 10.1007/BF02142692. [15] Y. Censor, T. Elfving, N. Kopf and T. Bortfeld, The multiple-sets split feasibility problem and its applications, Inverse Problems, 21 (2005), 2071-2084.  doi: 10.1088/0266-5611/21/6/017. [16] Y. Censor, A. Gibali, S. Reich and S. Sabach, Common solutions to variational inequalities, Set-Valued Var. Anal., 20 (2012), 229-247.  doi: 10.1007/s11228-011-0192-x. [17] H. G. Chen and R. T. Rockafellar, Convergence rates in forward-backward splitting, SIAM J. Optim., 7 (1997), 421-444.  doi: 10.1137/S1052623495290179. [18] S. Chen, D. Donoho and M. Saunders, Atomic decomposition by basis pursuit, SIAM J. Sci. Comput., 20 (1998), 33-61.  doi: 10.1137/S1064827596304010. [19] P. L. Combettes and V. R. Wajs, Signal recovery by proximal forward-backward splitting, Multiscale Model. Simul., 4 (2005), 1168-1200.  doi: 10.1137/050626090. [20] I. Daubechies, M. Defrise and C. De Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Commun. Pure Appl. Math., 57 (2004), 1413-1457.  doi: 10.1002/cpa.20042. [21] Q. L. Dong, Y. J. Cho, L. L. Zhong and T. M. Rassias, Inertial projection and contraction algorithms for variational inequalities, J. Global Optim., 70 (2018), 687-704.  doi: 10.1007/s10898-017-0506-0. [22] J. Duchi and Y. Singer, Efficient online and batch learning using forward-backward splitting, J. Mach. Learn. Res., 10 (2009), 2899-2934. [23] M. Eslamian, Strong convergence theorem for common zero points of inverse strongly monotone mappings and common fixed points of generalized demimetric mappings, Optimization, 2021. doi: 10.1080/02331934.2021.1939341. [24] M. Eslamian, Y. Shehu and O. S. Iyiola, A novel iterative algorithm with convergence analysis for split common fixed points and variational inequality prblems, Fixed Point Theory, 22 (2021), 123-140.  doi: 10.24193/fpt-ro.2021.1.09. [25] A. Gibali and D. V. Thong, Tseng type methods for solving inclusion problems and its applications, Calcolo, 55 (2018), Article ID 49. doi: 10.1007/s10092-018-0292-1. [26] R. Glowinski, J. L. Lions and R. Tremolieres, Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, 1981. [27] S. He and C. Yang, Solving the variational inequality problem defined on intersectoin of finite level sets, Abstr. Appl. Anal., 2013 (2013), Article ID 942315, 8pp. doi: 10.1155/2013/942315. [28] T. Kawasaki and W. Takahashi, A strong convergence theorem for countable families of nonlinear nonself mappings in Hilbert spaces and applications, J. Nonlinear Convex Anal., 19 (2018), 543-560. [29] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980. [30] D. Kitkuan, P. Kumam, J. Martínez-Moreno and K. Sitthithakerngkiet, Inertial viscosity forward-backward splitting algorithm for monotone inclusions and its application to image restoration problems, Int. J. Comput. Math., 97 (2019), 482-497.  doi: 10.1080/00207160.2019.1649661. [31] P. L. Lions and B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM J. Numer. Anal., 16 (1979), 964-979.  doi: 10.1137/0716071. [32] D. A. Lorenz and T. Pock, An inertial forward-backward algorithm for monotone inclusions, J. Math. Imaging Vision, 51 (2015), 311-325.  doi: 10.1007/s10851-014-0523-2. [33] E. Masad and S. Reich, A note on the multiple-set split convex feasibility problem, J. Nonlinear Convex Anal., 8 (2007), 367-371. [34] A. Moudafi, Vicosity approximation methods for fixed point problems, J. Math. Anal. Appl., 241 (2000), 46-55.  doi: 10.1006/jmaa.1999.6615. [35] Y. Nesterov, A method for solving the convex programming problem with convergence rate O(1/k2), Dokl. Akad. Nauk SSSR, 269 (1983), 543-547. [36] G. N. Ogwo, T. O. Alakoya and O. T. Mewomo, Iterative algorithm with self-adaptive step size for approximating the common solution of variational inequality and fixed point problems, Optimization, 2021. doi: 10.1080/02331934.2021.1981897. [37] G. N. Ogwo, C. Izuchukwu and O. T. Mewomo, Inertial methods for finding minimumnorm solutions of the split variational inequality problem beyond monotonicity, Numer. Algorithms, 88 (2021), 1419-1456.  doi: 10.1007/s11075-021-01081-1. [38] M. A. Olona, T. O. Alakoya, A. O.-E. Owolabi and O. T. Mewomo, Inertial shrinking projection algorithm with self-adaptive step size for split generalized equilibrium and fixed point problems for a countable family of nonexpansive multivalued mappings, Demonstr. Math., 54 (2021), 47-67.  doi: 10.1515/dema-2021-0006. [39] A. O. E. Owolabi, T. O. Alakoya, A. Taiwo and O. T. Mewomo, A new inertial-projection algorithm for approximating common solution of variational inequality and fixed point problems of multivalued mappings, Numer. Algebra Control Optim., 2021 doi: 10.3934/naco.2021004. [40] N. Parikh and S. Boyd, Proximal Algorithms, Found. Trends Optim., 2014. doi: 10.1561/9781601987174. [41] G. B. Passty, Ergodic convergence to a zero of the sum of monotone operators in Hilbert spaces, J. Math. Anal. Appl., 72 (1979), 383-390.  doi: 10.1016/0022-247X(79)90234-8. [42] B. T. Polyak, Some methods of speeding up the convergence of iteration methods, Ž. Vyčisl. Mat i Mat. Fiz., 4 (1964), 791-803. [43] H. Raguet, J. Fadili and G. Peyré, A generalized forward-backward splitting, SIAM J. Imaging Sci., 6 (2013), 1199-1226.  doi: 10.1137/120872802. [44] R. T. Rockafellar, Characterization of the subdifferentials of convex functions, Pacific J. Math., 17 (1966), 497-510.  doi: 10.2140/pjm.1966.17.497. [45] R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), 877-898.  doi: 10.1137/0314056. [46] W. Takahashi, Nonlinear Functional Analysis-Fixed Point Theory and Its Applications, Yokohama Publishers, Yokohama, 2000. [47] W. Takahashi, Strong convergence theorem for a finite family of demimetric mappings with variational inequality problems in a Hilbert space, Jpn. J. Ind. Appl. Math., 34 (2017), 41-57.  doi: 10.1007/s13160-017-0237-0. [48] W. Takahashi, The split common fixed point problem and the shrinking projection method in Banach spaces, J. Convex Anal., 24 (2017), 1015-1028. [49] W. Takahashi, Weak and strong convergence theorems for new demimetric mappings and the split common fixed point problem in Banach spaces, Numer. Funct. Anal. Appl. Optim., 39 (2018), 1011-1033.  doi: 10.1080/01630563.2018.1466803. [50] B. Tan and S. Y. Cho, Strong convergence of inertial forward backward methods for solving monotone inclusions, Applicable Analysis, 2021. doi: 10.1080/00036811.2021.1892080. [51] B. Tan, X. Qin and J. C. Yao, Strong convergence of self-adaptive inertial algorithms for solving split variational inclusion problems with applications, J. Sci. Comput., 87 (2021), Article ID 20, 34pp. doi: 10.1007/s10915-021-01428-9. [52] M. Tian and G. Xu, Inertial modified projection algorithm with self-adaptive technique for solving pseudo-monotone variational inequality problems in Hilbert spaces, Optimization, 2021. doi: 10.1080/02331934.2021.1928123. [53] R. Tibshirami, Regression shrinkage and selection via lasso, J. Roy. Statist. Soc. Ser. B, 58 (1996), 267-288.  doi: 10.1111/j.2517-6161.1996.tb02080.x. [54] P. Tseng, A modified forward backward splitting method for maximal monotone mappings, SIAM J. Control Optim., 38 (2000), 431-446.  doi: 10.1137/S0363012998338806. [55] E. Zeidler, Nonlinear Functional Analysis and Its Applications Ⅲ: Variational Methods and Optimization, Springer-Verlag, New York, 1985. [56] T. Y. Zhao, D. Q. Wang and L. C. Ceng et al., Inertial method for split null point problems with pseudomonotone variational inequality problems, Numer. Funct. Anal. Appl., 42 (2020), 69-90.
The graph of the error $\|x_n-x_{n-1}\|_2$
The graph of $x_n$
The graph of the value of $\frac{1}{2} \|Tx_n-b\|^2$ for the algorithms
Comparison of the new algorithm and the Algorithm 1 to recovery of a sparse signal with 5% nonzero elements
The results of the new algorithm for Example 5.1
 Starting points CPU time No. iterations $x_0=t^2$ $x_1=sin(2t)$ 0.62 s 1 $x_0=t$ $x_1=3e^{-2t}$ 1.42 s 2 $x_0=10\;\; t\;\; sin(5t)$ $x_1=10\;\; t\;\; sin(5t)$ 0.61 s 1
 Starting points CPU time No. iterations $x_0=t^2$ $x_1=sin(2t)$ 0.62 s 1 $x_0=t$ $x_1=3e^{-2t}$ 1.42 s 2 $x_0=10\;\; t\;\; sin(5t)$ $x_1=10\;\; t\;\; sin(5t)$ 0.61 s 1
Numerical results of comparison of the new algorithm and the Algorithm 1 for Example 5.2
 Starting point(s) Algorithm 1 The new algorithm No. iterations CPU time No. iterations CPU time $[-5,4,-3,2,-1]$ 63 0.0133 31 0.0025 $[-50,40,-30,20,-1]$ 72 0.0263 35 0.0093 $[5,4,3,2,1]$ 64 0.0146 31 0.0120 $[50,40,30,20,10]$ 72 0.0150 29 0.0095
 Starting point(s) Algorithm 1 The new algorithm No. iterations CPU time No. iterations CPU time $[-5,4,-3,2,-1]$ 63 0.0133 31 0.0025 $[-50,40,-30,20,-1]$ 72 0.0263 35 0.0093 $[5,4,3,2,1]$ 64 0.0146 31 0.0120 $[50,40,30,20,10]$ 72 0.0150 29 0.0095
The details of iterations of the new algorithm for Example 5.2
 n $x_n$ $\|x_n-x_{n-1}\|_2$ 0 [-5.000, 4.0000, -3.0000, 2.0000, -1.0000] — 1 [-5.000, 4.0000, -3.0000, 2.0000, -1.0000] — 2 [-6.3896, 5.0620, -4.2310, 2.4069, -1.5759] 2.2520e+00 8 [-0.0295, 0.0190, -0.0547, -0.0021, -0.0336] 1.8203e-01 14 [ 0.0019, -0.0014, 0.0024, -0.0003, 0.0013] 2.6378e-03 20 [ 0.0000, -0.0000, 0.0000, -0.0000, 0.0000] 5.5509e-05 22 [ 0.0000, -0.0000, 0.0000, -0.0000, 0.0000] 8.9023e-06
 n $x_n$ $\|x_n-x_{n-1}\|_2$ 0 [-5.000, 4.0000, -3.0000, 2.0000, -1.0000] — 1 [-5.000, 4.0000, -3.0000, 2.0000, -1.0000] — 2 [-6.3896, 5.0620, -4.2310, 2.4069, -1.5759] 2.2520e+00 8 [-0.0295, 0.0190, -0.0547, -0.0021, -0.0336] 1.8203e-01 14 [ 0.0019, -0.0014, 0.0024, -0.0003, 0.0013] 2.6378e-03 20 [ 0.0000, -0.0000, 0.0000, -0.0000, 0.0000] 5.5509e-05 22 [ 0.0000, -0.0000, 0.0000, -0.0000, 0.0000] 8.9023e-06
Numerical results of comparison of the new algorithm and The Algorithm 1 to recovery of a sparse signal with $p$ % nonzero elements
 p Algorithm 1 New Algorithm CPU time No. iterations CPU time No. iterations 2 % 1.8809 s 606 0.3698 s 129 5 % 2.2180 s 790 0.4061 s 142 10 % 1.1052 s 374 0.5425 s 174
 p Algorithm 1 New Algorithm CPU time No. iterations CPU time No. iterations 2 % 1.8809 s 606 0.3698 s 129 5 % 2.2180 s 790 0.4061 s 142 10 % 1.1052 s 374 0.5425 s 174
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