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doi: 10.3934/jimo.2021095
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Inertial Tseng's extragradient method for solving variational inequality problems of pseudo-monotone and non-Lipschitz operators

1. 

School of Mathematics Science, Chongqing Normal University, Chongqing 401331, China

2. 

Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China

3. 

Department of Mathematics and Physical Sciences, , California University of Pennsylvania, PA, USA

* Corresponding author: G. Cai

Received  September 2020 Revised  January 2021 Early access May 2021

Fund Project: The first author is supported by the NSF of China (Grant No. 11771063), the Natural Science Foundation of Chongqing(cstc2020jcyj-msxmX0455), Science and Technology Project of Chongqing Education Committee (Grant No. KJZD-K201900504) and the Program of Chongqing Innovation Research Group Project in University (Grant no. CXQT19018)

In this paper, we propose a new inertial Tseng's extragradient iterative algorithm for solving variational inequality problems of pseudo-monotone and non-Lipschitz operator in real Hilbert spaces. We prove that the sequence generated by proposed algorithm converges strongly to an element of solutions of variational inequality problem under some suitable assumptions imposed on the parameters. Finally, we give some numerical experiments for supporting our main results. The main results obtained in this paper extend and improve some related works in the literature.

Citation: Gang Cai, Yekini Shehu, Olaniyi S. Iyiola. Inertial Tseng's extragradient method for solving variational inequality problems of pseudo-monotone and non-Lipschitz operators. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021095
References:
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T. O. AlakoyaL. O. Jolaoso and O. T. Mewomo, Modified inertial subgradient extragradient method with self adaptive stepsize for solving monotone variational inequality and fixed point problems, Optimization, 70 (2021), 545-574.  doi: 10.1080/02331934.2020.1723586.  Google Scholar

[2]

F. Alvarez and H. Attouch, An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping, Set. Valued Anal., 9 (2001), 3-11.  doi: 10.1023/A:1011253113155.  Google Scholar

[3]

F. Alvarez, Weak convergence of a relaxed and inertial hybrid projection proximal point algorithm for maximal monotone operators in Hilbert space, SIAM J. Optim., 14 (2004), 773-782.  doi: 10.1137/S1052623403427859.  Google Scholar

[4]

C. Baiocchi and A. Capelo, Variational and quasivariational inequalities: Applications to free boundary problems, Wiley, New York, 1984.  Google Scholar

[5]

R. I. BǫtE. R. Csetnek and A. Heinrich, A primal-dual splitting algorithm for finding zeros of sums of maximally monotone operators, SIAM J. Optim., 23 (2013), 2011-2036.  doi: 10.1137/12088255X.  Google Scholar

[6]

R. I. Bǫt and C. Hendrich, A Douglas-Rachford type primal-dual method for solving inclusions with mixtures of composite and parallel-sum type monotone operators, SIAM J. Optim., 23 (2013), 2541-2565.  doi: 10.1137/120901106.  Google Scholar

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R. I. BǫtE. R. CsetnekA. Heinrich and C. Hendrich, On the convergence rate improvement of a primal-dual splitting algorithm for solving monotone inclusion problems, Math. Program., 150 (2015), 251-279.  doi: 10.1007/s10107-014-0766-0.  Google Scholar

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R. I. BǫtE. R. Csetnek and C. Hendrich, Inertial Douglas-Rachford splitting for monotone inclusion problems, Appl. Math. Comput., 256 (2015), 472-487.  doi: 10.1016/j.amc.2015.01.017.  Google Scholar

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R. I. Bǫt and E. R. Csetnek, An inertial Tseng's type proximal algorithm for nonsmooth and nonconvex optimization problems, J. Optim. Theory Appl., 171 (2016), 600-616.  doi: 10.1007/s10957-015-0730-z.  Google Scholar

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R. I. BǫtE. R. Csetnek and P. T. Vuong, The forward-backward-forward method from continuous and discrete perspective for pseudo-monotone variational inequalities in Hilbert spaces, European J. Oper. Res., 287 (2020), 49-60.  doi: 10.1016/j.ejor.2020.04.035.  Google Scholar

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Y. CensorA. Gibali and S. Reich, The subgradient extragradient method for solving variational inequalities in Hilbert space, J. Optim. Theory Appl., 148 (2011), 318-335.  doi: 10.1007/s10957-010-9757-3.  Google Scholar

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Y. CensorA. Gibali and S. Reich, Extensions of Korpelevich's extragradient method for the variational inequality problem in Euclidean space, Optimization, 61 (2011), 1119-1132.  doi: 10.1080/02331934.2010.539689.  Google Scholar

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Y. CensorA. Gibali and S. Reich, Algorithms for the split variational inequality problem, Numer. Algorithms, 56 (2012), 301-323.  doi: 10.1007/s11075-011-9490-5.  Google Scholar

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S. V. DenisovV. V. Semenov and L. M. Chabak, Convergence of the modified extragradient method for variational inequalities with non-Lipschitz operators, Cybern. Syst. Anal., 51 (2015), 757-765.  doi: 10.1007/s10559-015-9768-z.  Google Scholar

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Q. L. DongY. J. ChoL. L. Zhong and Th. M. Rassias, Inertial projection and contraction algorithms for variational inequalities, J. Glob. Optim., 70 (2018), 687-704.  doi: 10.1007/s10898-017-0506-0.  Google Scholar

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Q. L. DongH. B.YuanY. J. Cho and Th. M. Rassias, Modified inertial Mann algorithm and inertial CQ-algorithm for nonexpansive mappings, Optim. Lett., 12 (2018), 87-102.  doi: 10.1007/s11590-016-1102-9.  Google Scholar

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Q.-L. Dong, K. R. Kazmi, R. Ali and X.-H. Li, Inertial Krasnosel'skii-Mann type hybrid algorithms for solving hierarchical fixed point problems, J. Fixed Point Theory Appl., 21 (2019), 57. doi: 10.1007/s11784-019-0699-6.  Google Scholar

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F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer Series in Operations Research, vol. II. Springer, New York, 2003.  Google Scholar

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A. GibaliD. V. Thong and P. A. Tuan, Two simple projection-type methods for solving variational inequalities, Anal. Math. Phys., 9 (2019), 2203-2225.  doi: 10.1007/s13324-019-00330-w.  Google Scholar

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P. T. Harker and J.-S. Pang, A damped-newton method for the linear complementarity problem, Lect. Appl. Math., 26 (1990), 265-284.  doi: 10.1007/bf01582255.  Google Scholar

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X. Hu and J. Wang, Solving pseudo-monotone variational inequalities and pseudo-convex optimization problems using the projection neural network, IEEE Trans. Neural Netw., 17 (2006), 1487-1499.   Google Scholar

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A. N. Iusem, An iterative algorithm for the variational inequality problem, Mat. Apl. Comput., 13 (1994), 103-114.   Google Scholar

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A. Iusem and R. Gárciga Otero, Inexact versions of proximal point and augmented Lagrangian algorithms in Banach spaces, Numer. Funct. Anal. Optim., 22 (2001), 609-640.  doi: 10.1081/NFA-100105310.  Google Scholar

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C. Izuchukwu, A. A. Mebawondu and O. T. Mewomo, A new method for solving split variational inequality problem without co-coerciveness, J. Fixed Point Theory Appl., 22 (2020), 98. doi: 10.1007/s11784-020-00834-0.  Google Scholar

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L. O. JolaosoA. TaiwoT. O. Alakoya and O. T. Mewomo, A strong convergence theorem for solving pseudo-monotone variational inequalities using projection methods, J. Optim. Theory Appl., 185 (2020), 744-766.  doi: 10.1007/s10957-020-01672-3.  Google Scholar

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L. O. Jolaoso, A. Taiwo, T. O. Alakoya and O. T. Mewomo, A unified algorithm for solving variational inequality and fixed point problems with application to the split equality problem, Comput. Appl. Math., 39 (2020), 38. doi: 10.1007/s40314-019-1014-2.  Google Scholar

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L. O. JolaosoT. O. AlakoyaA. Taiwo and O. T. Mewomo, Inertial extragradient method via viscosity approximation approach for solving Equilibrium problem in Hilbert space, Optimization, 70 (2021), 387-412.  doi: 10.1080/02331934.2020.1716752.  Google Scholar

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G. KassayS. Reich and S. Sabach, Iterative methods for solving systems of variational inequalities in refelexive Banach spaces, SIAM J. Optim., 21 (2011), 1319-1344.  doi: 10.1137/110820002.  Google Scholar

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P.-E. Maingé, Convergence theorem for inertial KM-type algorithms, J. Comput. Appl. Math., 219 (2008), 223-236.  doi: 10.1016/j.cam.2007.07.021.  Google Scholar

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Y. V. Malitsky and V. V. Semenov, A hybrid method without extrapolation step for solving variational inequality problems, J. Glob. Optim., 61 (2015), 193-202.  doi: 10.1007/s10898-014-0150-x.  Google Scholar

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Y. ShehuO. S. Iyiola and F. U. Ogbuisi, Iterative method with inertial terms for nonexpansive mappings: Applications to compressed sensing, Numer Algorithms, 83 (2020), 1321-1347.  doi: 10.1007/s11075-019-00727-5.  Google Scholar

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show all references

References:
[1]

T. O. AlakoyaL. O. Jolaoso and O. T. Mewomo, Modified inertial subgradient extragradient method with self adaptive stepsize for solving monotone variational inequality and fixed point problems, Optimization, 70 (2021), 545-574.  doi: 10.1080/02331934.2020.1723586.  Google Scholar

[2]

F. Alvarez and H. Attouch, An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping, Set. Valued Anal., 9 (2001), 3-11.  doi: 10.1023/A:1011253113155.  Google Scholar

[3]

F. Alvarez, Weak convergence of a relaxed and inertial hybrid projection proximal point algorithm for maximal monotone operators in Hilbert space, SIAM J. Optim., 14 (2004), 773-782.  doi: 10.1137/S1052623403427859.  Google Scholar

[4]

C. Baiocchi and A. Capelo, Variational and quasivariational inequalities: Applications to free boundary problems, Wiley, New York, 1984.  Google Scholar

[5]

R. I. BǫtE. R. Csetnek and A. Heinrich, A primal-dual splitting algorithm for finding zeros of sums of maximally monotone operators, SIAM J. Optim., 23 (2013), 2011-2036.  doi: 10.1137/12088255X.  Google Scholar

[6]

R. I. Bǫt and C. Hendrich, A Douglas-Rachford type primal-dual method for solving inclusions with mixtures of composite and parallel-sum type monotone operators, SIAM J. Optim., 23 (2013), 2541-2565.  doi: 10.1137/120901106.  Google Scholar

[7]

R. I. BǫtE. R. CsetnekA. Heinrich and C. Hendrich, On the convergence rate improvement of a primal-dual splitting algorithm for solving monotone inclusion problems, Math. Program., 150 (2015), 251-279.  doi: 10.1007/s10107-014-0766-0.  Google Scholar

[8]

R. I. BǫtE. R. Csetnek and C. Hendrich, Inertial Douglas-Rachford splitting for monotone inclusion problems, Appl. Math. Comput., 256 (2015), 472-487.  doi: 10.1016/j.amc.2015.01.017.  Google Scholar

[9]

R. I. Bǫt and E. R. Csetnek, An inertial Tseng's type proximal algorithm for nonsmooth and nonconvex optimization problems, J. Optim. Theory Appl., 171 (2016), 600-616.  doi: 10.1007/s10957-015-0730-z.  Google Scholar

[10]

R. I. BǫtE. R. Csetnek and P. T. Vuong, The forward-backward-forward method from continuous and discrete perspective for pseudo-monotone variational inequalities in Hilbert spaces, European J. Oper. Res., 287 (2020), 49-60.  doi: 10.1016/j.ejor.2020.04.035.  Google Scholar

[11]

Y. CensorA. Gibali and S. Reich, The subgradient extragradient method for solving variational inequalities in Hilbert space, J. Optim. Theory Appl., 148 (2011), 318-335.  doi: 10.1007/s10957-010-9757-3.  Google Scholar

[12]

Y. CensorA. Gibali and S. Reich, Extensions of Korpelevich's extragradient method for the variational inequality problem in Euclidean space, Optimization, 61 (2011), 1119-1132.  doi: 10.1080/02331934.2010.539689.  Google Scholar

[13]

Y. CensorA. Gibali and S. Reich, Algorithms for the split variational inequality problem, Numer. Algorithms, 56 (2012), 301-323.  doi: 10.1007/s11075-011-9490-5.  Google Scholar

[14]

R. W. Cottle and J. C. Yao, Pseudo-monotone complementarity problems in Hilbert space, J. Optim. Theory Appl., 75 (1992), 281-295.  doi: 10.1007/BF00941468.  Google Scholar

[15]

S. V. DenisovV. V. Semenov and L. M. Chabak, Convergence of the modified extragradient method for variational inequalities with non-Lipschitz operators, Cybern. Syst. Anal., 51 (2015), 757-765.  doi: 10.1007/s10559-015-9768-z.  Google Scholar

[16]

Q. L. DongY. J. ChoL. L. Zhong and Th. M. Rassias, Inertial projection and contraction algorithms for variational inequalities, J. Glob. Optim., 70 (2018), 687-704.  doi: 10.1007/s10898-017-0506-0.  Google Scholar

[17]

Q. L. DongH. B.YuanY. J. Cho and Th. M. Rassias, Modified inertial Mann algorithm and inertial CQ-algorithm for nonexpansive mappings, Optim. Lett., 12 (2018), 87-102.  doi: 10.1007/s11590-016-1102-9.  Google Scholar

[18]

Q.-L. Dong, K. R. Kazmi, R. Ali and X.-H. Li, Inertial Krasnosel'skii-Mann type hybrid algorithms for solving hierarchical fixed point problems, J. Fixed Point Theory Appl., 21 (2019), 57. doi: 10.1007/s11784-019-0699-6.  Google Scholar

[19]

F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer Series in Operations Research, vol. II. Springer, New York, 2003.  Google Scholar

[20]

A. GibaliD. V. Thong and P. A. Tuan, Two simple projection-type methods for solving variational inequalities, Anal. Math. Phys., 9 (2019), 2203-2225.  doi: 10.1007/s13324-019-00330-w.  Google Scholar

[21]

R. Glowinski, J.-L. Lions and R. Trémolières, Numerical Analysis of Variational Inequalities, Elsevier, Amsterdam, 1981.  Google Scholar

[22]

K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, New York, 1984.  Google Scholar

[23]

P. T. Harker and J.-S. Pang, A damped-newton method for the linear complementarity problem, Lect. Appl. Math., 26 (1990), 265-284.  doi: 10.1007/bf01582255.  Google Scholar

[24]

X. Hu and J. Wang, Solving pseudo-monotone variational inequalities and pseudo-convex optimization problems using the projection neural network, IEEE Trans. Neural Netw., 17 (2006), 1487-1499.   Google Scholar

[25]

A. N. Iusem, An iterative algorithm for the variational inequality problem, Mat. Apl. Comput., 13 (1994), 103-114.   Google Scholar

[26]

A. Iusem and R. Gárciga Otero, Inexact versions of proximal point and augmented Lagrangian algorithms in Banach spaces, Numer. Funct. Anal. Optim., 22 (2001), 609-640.  doi: 10.1081/NFA-100105310.  Google Scholar

[27]

C. Izuchukwu, A. A. Mebawondu and O. T. Mewomo, A new method for solving split variational inequality problem without co-coerciveness, J. Fixed Point Theory Appl., 22 (2020), 98. doi: 10.1007/s11784-020-00834-0.  Google Scholar

[28]

L. O. JolaosoA. TaiwoT. O. Alakoya and O. T. Mewomo, A strong convergence theorem for solving pseudo-monotone variational inequalities using projection methods, J. Optim. Theory Appl., 185 (2020), 744-766.  doi: 10.1007/s10957-020-01672-3.  Google Scholar

[29]

L. O. Jolaoso, A. Taiwo, T. O. Alakoya and O. T. Mewomo, A unified algorithm for solving variational inequality and fixed point problems with application to the split equality problem, Comput. Appl. Math., 39 (2020), 38. doi: 10.1007/s40314-019-1014-2.  Google Scholar

[30]

L. O. JolaosoT. O. AlakoyaA. Taiwo and O. T. Mewomo, Inertial extragradient method via viscosity approximation approach for solving Equilibrium problem in Hilbert space, Optimization, 70 (2021), 387-412.  doi: 10.1080/02331934.2020.1716752.  Google Scholar

[31]

G. KassayS. Reich and S. Sabach, Iterative methods for solving systems of variational inequalities in refelexive Banach spaces, SIAM J. Optim., 21 (2011), 1319-1344.  doi: 10.1137/110820002.  Google Scholar

[32] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and their Applications, Academic Press, New York, 1980.   Google Scholar
[33]

I. Konnov, Combined Relaxation Methods for Variational Inequalities, Springer, Berlin, 2001. doi: 10.1007/978-3-642-56886-2.  Google Scholar

[34]

G. M. Korpelevič, The extragradient method for finding saddle points and other problems, Ekon. Mat. Metody, 12 (1976), 747-756.   Google Scholar

[35]

P.-E. Maingé, A hybrid extragradient-viscosity method for monotone operators and fixed point problems, SIAM J. Control Optim., 47 (2008), 1499-1515.  doi: 10.1137/060675319.  Google Scholar

[36]

P.-E. Maingé, Convergence theorem for inertial KM-type algorithms, J. Comput. Appl. Math., 219 (2008), 223-236.  doi: 10.1016/j.cam.2007.07.021.  Google Scholar

[37]

Y. Malitsky, Projected reflected gradient methods for monotone variational inequalities, SIAM J. Optim., 25 (2015), 502-520.  doi: 10.1137/14097238X.  Google Scholar

[38]

Y. V. Malitsky and V. V. Semenov, A hybrid method without extrapolation step for solving variational inequality problems, J. Glob. Optim., 61 (2015), 193-202.  doi: 10.1007/s10898-014-0150-x.  Google Scholar

[39]

Y. Shehu and O. S. Iyiola, Convergence analysis for the proximal split feasibility problem using an inertial extrapolation term method, J. Fixed Point Theory Appl., 19 (2017), 2483-2510.  doi: 10.1007/s11784-017-0435-z.  Google Scholar

[40]

Y. ShehuO. S. Iyiola and F. U. Ogbuisi, Iterative method with inertial terms for nonexpansive mappings: Applications to compressed sensing, Numer Algorithms, 83 (2020), 1321-1347.  doi: 10.1007/s11075-019-00727-5.  Google Scholar

[41]

Y. Shehu and P. Cholamjiak, Iterative method with inertial for variational inequalities in Hilbert spaces, Calcolo, 56 (2019), 4. doi: 10.1007/s10092-018-0300-5.  Google Scholar

[42]

Y. ShuhuQ.-L. Dong and D. Jiang, Single projection method for pseudo-monotone variational inequality in Hilbert spaces, Optimization, 68 (2019), 385-409.  doi: 10.1080/02331934.2018.1522636.  Google Scholar

[43]

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Figure 1.  Example 1: $ k = 20 $, $ N = 10 $
Figure 2.  Example 1: $ k = 20 $, $ N = 20 $
Figure 3.  Example 1: $ k = 20 $, $ N = 30 $
Figure 4.  Example 1: $ k = 20 $, $ N = 40 $
Figure 5.  Example 1: $ k = 30 $, $ N = 10 $
Figure 6.  Example 1: $ k = 30 $, $ N = 20 $
Figure 7.  Example 1: $ k = 30 $, $ N = 30 $
Figure 8.  Example 1: $ k = 30 $, $ N = 40 $
Figure 9.  Example 1: Different $ \gamma $ with $ (N, k) = (20, 10) $
Figure 10.  Example 1: Different $ \gamma $ with $ (N, k) = (20, 21) $
Figure 11.  Example 1: Different $ \gamma $ with $ (N, k) = (20, 30) $
Figure 12.  Example 1: Different $ \gamma $ with $ (N, k) = (20, 40) $
Figure 13.  Example 1: Different $ \gamma $ with $ (N, k) = (30, 10) $
Figure 14.  Example 1: Different $ \gamma $ with $ (N, k) = (30, 20) $
Figure 15.  Example 1: Different $ \gamma $ with $ (N, k) = (30, 30) $
Figure 16.  Example 1: Different $ \gamma $ with $ (N, k) = (30, 40) $
Figure 17.  Example 1: Different $ \mu $ with $ (N, k) = (20, 10) $
Figure 18.  Example 1: Different $ \mu $ with $ (N, k) = (20, 20) $
Figure 19.  Example 1: Different $ \mu $ with $ (N, k) = (20, 30) $
Figure 20.  Example 1: Different $ \mu $ with $ (N, k) = (20, 40) $
Figure 21.  Example 1: Different $ \mu $ with $ (N, k) = (30, 10) $
Figure 22.  Example 1: Different $ \mu $ with $ (N, k) = (30, 20) $
Figure 23.  Example 1: Different $ \mu $ with $ (N, k) = (30, 30) $
Figure 24.  Example 1: Different $ \mu $ with $ (N, k) = (30, 40) $
Figure 25.  Example 2: Case I
Figure 26.  Example 2: Case II
Figure 27.  Example 2: Case III
Figure 28.  Example 2: Case IV
Figure 29.  Example 2: Case V
Figure 30.  Example 2: Case VI
Figure 31.  Example 2: Case I with different $ \gamma $
Figure 32.  Example 2: Case II with different $ \gamma $
Figure 33.  Example 2: Case III with different $ \gamma $
Figure 34.  Example 2: Case IV with different $ \gamma $
Figure 35.  Example 2: Case V with different $ \gamma $
Figure 36.  Example 2: Case VI with different $ \gamma $
Figure 37.  Example 2: Case I with different $ \mu $
Figure 38.  Example 2: Case II with different $ \mu $
Figure 39.  Example 2: Case III with different $ \mu $
Figure 40.  Example 2: Case IV with different $ \mu $
Figure 41.  Example 2: Case V with different $ \mu $
Figure 42.  Example 2: Case VI with different $ \mu $
Figure 43.  The value of error versus the iteration numbers for Example 3
Table 1.  Methods Parameters Choice for Comparison
Proposed Alg. $ \epsilon_n = \frac{1}{n^2} $ $ \theta = 0.1 $ $ l=0.001 $ $ \beta_n = \frac{1}{n} $
$ \gamma = 0.99 $ $ \mu = 0.99 $
Thong Alg. (1) $ \epsilon_n = \frac{1}{(n + 1)^2} $ $ \theta = 0.1 $ $ \beta_n = \frac{1}{n + 1} $ $ \lambda = \frac{1}{1.01L} $
Thong Alg. (2) $ l=0.001 $ $ \gamma = 0.99 $ $ \mu = 0.99 $
Thong Alg. (3) $ \alpha_n = \frac{1}{n + 1} $ $ l=0.001 $ $ \gamma = 0.99 $ $ \mu = 0.99 $
Gibali Alg. $ \alpha_n = \frac{1}{n + 1} $ $ l=0.001 $ $ \gamma = 0.99 $ $ \mu = 0.99 $
Proposed Alg. $ \epsilon_n = \frac{1}{n^2} $ $ \theta = 0.1 $ $ l=0.001 $ $ \beta_n = \frac{1}{n} $
$ \gamma = 0.99 $ $ \mu = 0.99 $
Thong Alg. (1) $ \epsilon_n = \frac{1}{(n + 1)^2} $ $ \theta = 0.1 $ $ \beta_n = \frac{1}{n + 1} $ $ \lambda = \frac{1}{1.01L} $
Thong Alg. (2) $ l=0.001 $ $ \gamma = 0.99 $ $ \mu = 0.99 $
Thong Alg. (3) $ \alpha_n = \frac{1}{n + 1} $ $ l=0.001 $ $ \gamma = 0.99 $ $ \mu = 0.99 $
Gibali Alg. $ \alpha_n = \frac{1}{n + 1} $ $ l=0.001 $ $ \gamma = 0.99 $ $ \mu = 0.99 $
Table 2.  Example 1: Comparison among methods with different values of $ N $ and $ k $
$ N=10 $ $ N=20 $ $ N=30 $ $ N=40 $
$ k=20 $ Iter. Time Iter. Time Iter. Time Iter. Time
Proposed Alg. 3 1.3843 3 1.7672 3 1.7564 4 2.2017
Thong Alg. (1) 76 1.2902 139 2.7111 111 2.1715 232 37.7743
Thong Alg. (2) 2136 36.6812 1561 30.7776 1370 31.8672 1160 4.0453
Thong Alg. (3) 86 1.1655 152 2.3615 148 2.4878 178 29.0789
Gibali Alg. 150 12.0085 235 20.3243 319 41.0421 315 4.2520
$ N=10 $ $ N=20 $ $ N=30 $ $ N=40 $
$ k=30 $ Iter. Time Iter. Time Iter. Time Iter. Time
Proposed Alg. 3 1.3819 3 1.7834 3 1.6555 3 1.7517
Thong Alg. (1) 72 1.1548 142 2.6436 136 2.888 207 4.4416
Thong Alg. (2) 1771 30.2921 1325 28.4023 1132 28.6053 920 26.5714
Thong Alg. (3) 101 1.5058 90 1.4923 156 2.9515 162 3.6149
Gibali Alg. 203 17.1568 255 30.849 282 31.6244 303 35.2953
$ N=10 $ $ N=20 $ $ N=30 $ $ N=40 $
$ k=20 $ Iter. Time Iter. Time Iter. Time Iter. Time
Proposed Alg. 3 1.3843 3 1.7672 3 1.7564 4 2.2017
Thong Alg. (1) 76 1.2902 139 2.7111 111 2.1715 232 37.7743
Thong Alg. (2) 2136 36.6812 1561 30.7776 1370 31.8672 1160 4.0453
Thong Alg. (3) 86 1.1655 152 2.3615 148 2.4878 178 29.0789
Gibali Alg. 150 12.0085 235 20.3243 319 41.0421 315 4.2520
$ N=10 $ $ N=20 $ $ N=30 $ $ N=40 $
$ k=30 $ Iter. Time Iter. Time Iter. Time Iter. Time
Proposed Alg. 3 1.3819 3 1.7834 3 1.6555 3 1.7517
Thong Alg. (1) 72 1.1548 142 2.6436 136 2.888 207 4.4416
Thong Alg. (2) 1771 30.2921 1325 28.4023 1132 28.6053 920 26.5714
Thong Alg. (3) 101 1.5058 90 1.4923 156 2.9515 162 3.6149
Gibali Alg. 203 17.1568 255 30.849 282 31.6244 303 35.2953
Table 3.  Example 1 Comparison: Proposed Alg. with different values $ \gamma $
$ (N, k) $ $ \gamma = 0.1 $ $ \gamma = 0.5 $ $ \gamma = 0.7 $ $ \gamma = 0.99 $
$ (20, 10) $ No. of Iterations 3 3 3 3
CPU (Time) 1.6337 1.4830 1.4773 1.3843
$ (20, 20) $ No. of Iterations 3 3 4 3
CPU (Time) 1.4606 1.4876 2.3980 1.7672
$ (20, 30) $ No. of Iterations 4 5 4 3
CPU (Time) 1.8664 0.85257 1.7597 1.7564
$ (20, 40) $ No. of Iterations 4 3 4 4
CPU (Time) 1.6573 1.5935 1.8008 2.2017
$ (30, 10) $ No. of Iterations 3 3 3 3
CPU (Time) 1.3060 1.3376 1.4359 1.3819
$ (30, 20) $ No. of Iterations 3 4 3 3
CPU (Time) 1.4630 1.7306 1.5115 1.7834
$ (30, 30) $ No. of Iterations 4 3 4 3
CPU (Time) 1.7102 1.6399 1.7931 1.6555
$ (30, 40) $ No. of Iterations 5 3 4 3
CPU (Time) 2.7099 1.6589 2.3287 1.7517
$ (N, k) $ $ \gamma = 0.1 $ $ \gamma = 0.5 $ $ \gamma = 0.7 $ $ \gamma = 0.99 $
$ (20, 10) $ No. of Iterations 3 3 3 3
CPU (Time) 1.6337 1.4830 1.4773 1.3843
$ (20, 20) $ No. of Iterations 3 3 4 3
CPU (Time) 1.4606 1.4876 2.3980 1.7672
$ (20, 30) $ No. of Iterations 4 5 4 3
CPU (Time) 1.8664 0.85257 1.7597 1.7564
$ (20, 40) $ No. of Iterations 4 3 4 4
CPU (Time) 1.6573 1.5935 1.8008 2.2017
$ (30, 10) $ No. of Iterations 3 3 3 3
CPU (Time) 1.3060 1.3376 1.4359 1.3819
$ (30, 20) $ No. of Iterations 3 4 3 3
CPU (Time) 1.4630 1.7306 1.5115 1.7834
$ (30, 30) $ No. of Iterations 4 3 4 3
CPU (Time) 1.7102 1.6399 1.7931 1.6555
$ (30, 40) $ No. of Iterations 5 3 4 3
CPU (Time) 2.7099 1.6589 2.3287 1.7517
Table 4.  Example 1 Comparison: Proposed Alg. with different values $ \mu $
$ (N, k) $ $ \mu = 0.1 $ $ \mu = 0.5 $ $ \mu = 0.7 $ $ \mu = 0.99 $
$ (20, 10) $ No. of Iterations 3 3 4 3
CPU (Time) 1.4409 1.4632 2.6888 1.3843
$ (20, 20) $ No. of Iterations 3 3 3 3
CPU (Time) 1.5248 1.4840 1.5217 1.7672
$ (20, 30) $ No. of Iterations 4 3 5 3
CPU (Time) 1.9571 1.5852 1.9322 1.7564
$ (20, 40) $ No. of Iterations 6 4 5 4
CPU (Time) 3.0365 1.8605 2.0718 2.2017
$ (30, 10) $ No. of Iterations 3 3 3 3
CPU (Time) 1.3524 1.3416 1.3648 1.3819
$ (30, 20) $ No. of Iterations 3 3 4 3
CPU (Time) 1.5265 1.5336 1.6929 1.7834
$ (30, 30) $ No. of Iterations 5 5 3 3
CPU (Time) 2.9525 2.0816 1.6424 1.6555
$ (30, 40) $ No. of Iterations 3 4 8 3
CPU (Time) 1.6958 2.0833 4.5199 1.7517
$ (N, k) $ $ \mu = 0.1 $ $ \mu = 0.5 $ $ \mu = 0.7 $ $ \mu = 0.99 $
$ (20, 10) $ No. of Iterations 3 3 4 3
CPU (Time) 1.4409 1.4632 2.6888 1.3843
$ (20, 20) $ No. of Iterations 3 3 3 3
CPU (Time) 1.5248 1.4840 1.5217 1.7672
$ (20, 30) $ No. of Iterations 4 3 5 3
CPU (Time) 1.9571 1.5852 1.9322 1.7564
$ (20, 40) $ No. of Iterations 6 4 5 4
CPU (Time) 3.0365 1.8605 2.0718 2.2017
$ (30, 10) $ No. of Iterations 3 3 3 3
CPU (Time) 1.3524 1.3416 1.3648 1.3819
$ (30, 20) $ No. of Iterations 3 3 4 3
CPU (Time) 1.5265 1.5336 1.6929 1.7834
$ (30, 30) $ No. of Iterations 5 5 3 3
CPU (Time) 2.9525 2.0816 1.6424 1.6555
$ (30, 40) $ No. of Iterations 3 4 8 3
CPU (Time) 1.6958 2.0833 4.5199 1.7517
Table 5.  Methods Parameters Choice for Comparison
Proposed Alg. $ \epsilon_n = \frac{1}{(n + 1)^2} $ $ \theta = 0.5 $ $ l=0.01 $ $ \beta_n = \frac{1}{n + 1} $ $ \gamma = 0.99 $ $ \mu = 0.99 $
Gibali Alg. $ \alpha_n = \frac{1}{1 + n} $ $ l=0.01 $ $ \gamma = 0.99 $ $ \mu = 0.99 $
Proposed Alg. $ \epsilon_n = \frac{1}{(n + 1)^2} $ $ \theta = 0.5 $ $ l=0.01 $ $ \beta_n = \frac{1}{n + 1} $ $ \gamma = 0.99 $ $ \mu = 0.99 $
Gibali Alg. $ \alpha_n = \frac{1}{1 + n} $ $ l=0.01 $ $ \gamma = 0.99 $ $ \mu = 0.99 $
Table 6.  Example 2: Prop. Alg. vs Gibali Alg. (Unaccel. Alg.)
No. of Iterations CPU Time
Prop. Alg. Gibali Alg. Prop. Alg. Gibali Alg.
Case I 17 1712 0.001243 0.1244
Case II 17 1708 0.001518 0.1248
Case III 17 1713 0.001261 0.1276
Case IV 17 1729 0.001202 0.1297
Case V 17 1715 0.001272 0.1258
Case VI 18 1835 0.001339 0.1564
No. of Iterations CPU Time
Prop. Alg. Gibali Alg. Prop. Alg. Gibali Alg.
Case I 17 1712 0.001243 0.1244
Case II 17 1708 0.001518 0.1248
Case III 17 1713 0.001261 0.1276
Case IV 17 1729 0.001202 0.1297
Case V 17 1715 0.001272 0.1258
Case VI 18 1835 0.001339 0.1564
Table 7.  Example 2 Comparison: Proposed Alg. with different values $ \mu $
$ \mu = 0.1 $ $ \mu = 0.5 $ $ \mu = 0.7 $ $ \mu = 0.99 $
Case I No. of Iterations 17 17 17 17
CPU (Time) 0.0011992 0.0012179 0.0013264 0.0012430
Case II No. of Iterations 17 17 17 17
CPU (Time) 0.0011457 0.0011586 0.0015604 0.0015181
Case III No. of Iterations 17 17 17 17
CPU (Time) 0.0011386 0.0014248 0.0012852 0.0012606
Case IV No. of Iterations 17 17 17 17
CPU (Time) 0.0010843 0.0010928 0.0011176 0.0012022
Case V No. of Iterations 17 17 17 17
CPU (Time) 0.0012491 0.0011169 0.0012293 0.0012719
Case VI No. of Iterations 18 18 18 18
CPU (Time) 0.0012431 0.0013496 0.0011613 0.0013392
$ \mu = 0.1 $ $ \mu = 0.5 $ $ \mu = 0.7 $ $ \mu = 0.99 $
Case I No. of Iterations 17 17 17 17
CPU (Time) 0.0011992 0.0012179 0.0013264 0.0012430
Case II No. of Iterations 17 17 17 17
CPU (Time) 0.0011457 0.0011586 0.0015604 0.0015181
Case III No. of Iterations 17 17 17 17
CPU (Time) 0.0011386 0.0014248 0.0012852 0.0012606
Case IV No. of Iterations 17 17 17 17
CPU (Time) 0.0010843 0.0010928 0.0011176 0.0012022
Case V No. of Iterations 17 17 17 17
CPU (Time) 0.0012491 0.0011169 0.0012293 0.0012719
Case VI No. of Iterations 18 18 18 18
CPU (Time) 0.0012431 0.0013496 0.0011613 0.0013392
Table 8.  Example 2 Comparison: Proposed Alg. with different values $ \gamma $
$ \gamma = 0.1 $ $ \gamma = 0.5 $ $ \gamma = 0.7 $ $ \gamma = 0.99 $
Case I No. of Iterations 17 17 17 17
CPU (Time) 0.0013518 0.0012097 0.0011754 0.0012430
Case II No. of Iterations 17 17 17 17
CPU (Time) 0.0012701 0.0011233 0.0012382 0.0015181
Case III No. of Iterations 17 17 17 17
CPU (Time) 0.0011386 0.0014248 0.0012852 0.0012606
Case IV No. of Iterations 17 17 17 17
CPU (Time) 0.0011530 0.0013917 0.0015395 0.0012022
Case V No. of Iterations 17 17 17 17
CPU (Time) 0.0011413 0.0011319 0.0011286 0.0012719
Case VI No. of Iterations 17 17 18 18
CPU (Time) 0.0011094 0.0011839 0.0013550 0.0013392
$ \gamma = 0.1 $ $ \gamma = 0.5 $ $ \gamma = 0.7 $ $ \gamma = 0.99 $
Case I No. of Iterations 17 17 17 17
CPU (Time) 0.0013518 0.0012097 0.0011754 0.0012430
Case II No. of Iterations 17 17 17 17
CPU (Time) 0.0012701 0.0011233 0.0012382 0.0015181
Case III No. of Iterations 17 17 17 17
CPU (Time) 0.0011386 0.0014248 0.0012852 0.0012606
Case IV No. of Iterations 17 17 17 17
CPU (Time) 0.0011530 0.0013917 0.0015395 0.0012022
Case V No. of Iterations 17 17 17 17
CPU (Time) 0.0011413 0.0011319 0.0011286 0.0012719
Case VI No. of Iterations 17 17 18 18
CPU (Time) 0.0011094 0.0011839 0.0013550 0.0013392
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