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doi: 10.3934/jimo.2020123

New inertial method for generalized split variational inclusion problems

1. 

Department of Mathematics, Faculty of Science, Burapha University, Chonburi 20131, Thailand

2. 

School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban South Africa

3. 

University of Nigeria, Department of Mathematics, Nsukka, Nigeria

4. 

Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, People's Republic of China

5. 

School of Science, University of Phayao, Phayao, Thailand

* Corresponding author: F. U. Ogbuisi

Received  December 2019 Revised  January 2020 Published  June 2020

Fund Project: The work of the second author is based on the research supported wholly by the National Research Foundation (NRF) of South Africa (Grant Numbers: 111992). Opinions expressed and conclusions arrived are those of the authors and are not necessarily to be attributed to the NRF. This work was supported by Thailand Science Research and Innovation grant no. IRN62W0007 and Thailand Research Fund and University of Phayao grant no. RSA6180084

The purpose of this paper is to introduce a new inertial iterative method for solving split variational inclusion problems in real Hilbert spaces. We prove that the generated sequence converges weakly to the solution of the considered problem under some mild conditions. The major contributions of our results are: (ⅰ) to increase the rate of convergence of the method for solving split variational inclusion problem through the inertial extrapolation step, (ⅱ) to relax the choice of the inertial factor and show the inertial factor can be chosen greater than 1/3 unlike what is previously known before for inertial proximal point method in the literature (ⅲ) to show the numerical efficiency and superiority of our proposed method through some test example.

Citation: Preeyanuch Chuasuk, Ferdinard Ogbuisi, Yekini Shehu, Prasit Cholamjiak. New inertial method for generalized split variational inclusion problems. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020123
References:
[1]

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

[2]

H. H. Bauschke and P. L. Combettes, A weak-to-strong convergence principle for Fej$\acute{e}$r monotone methods in Hilbert spaces, Math. Oper. Res., 26 (2001), 248-264.  doi: 10.1287/moor.26.2.248.10558.  Google Scholar

[3]

H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, New York, 2011. doi: 10.1007/978-1-4419-9467-7.  Google Scholar

[4]

H. Brézis and P.-L. Lions, Produits infinis de résolvantes, Israel J. Math., 29 (1978), 329-345.  doi: 10.1007/BF02761171.  Google Scholar

[5]

C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Probl., 18 (2002), 441-453.  doi: 10.1088/0266-5611/18/2/310.  Google Scholar

[6]

C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Probl., 20 (2004), 103-120.  doi: 10.1088/0266-5611/20/1/006.  Google Scholar

[7]

Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projection in a product space, Numer. Algorithms, 8 (1994), 221-239.  doi: 10.1007/BF02142692.  Google Scholar

[8]

Y. CensorT. BortfeldB. Martin and A. Trofimov, A unified approach for inversion problems in intensity modulated radiation therapy, Phys. Med. Biol., 51 (2003), 2353-2365.  doi: 10.1088/0031-9155/51/10/001.  Google Scholar

[9]

C.-S. Chuang, Strong convergence theorems for the split variational inclusion problem in Hilbert spaces, Fixed Point Theory and Appl., 2013 (2013), 20 pp. doi: 10.1186/1687-1812-2013-350.  Google Scholar

[10]

C.-S. Chuang, Simultaneous subgradient algorithms for the generalized split variational inclusion problem in Hilbert spaces, Numer. Funct. Anal. Optim., 38 (2017), 306-326.  doi: 10.1080/01630563.2016.1233120.  Google Scholar

[11]

C.-S. Chuang, Hybrid inertial proximal algorithm for the split variational inclusion problem in Hilbert spaces with applications, Optimization, 66 (2017), 777-792.  doi: 10.1080/02331934.2017.1306744.  Google Scholar

[12]

P. L. Combettes, The convex feasibility problem in image recovery, Advances in Imaging and Electron Physics, 95 (1996), 155-270.  doi: 10.1016/S1076-5670(08)70157-5.  Google Scholar

[13]

J. ContrerasM. Klusch and J. B. Krawczyk, Numerical solution to Nash-Cournot equilibria in coupled constraint electricity markets, IEEE Trans. Power Syst., 19 (2004), 195-206.  doi: 10.1109/TPWRS.2003.820692.  Google Scholar

[14]

Y. Z. Dang and Y. Gao, The strong convergence of a KM-CQ-like algorithm for a split feasibility problem, Inverse Probl., 27 (2011), 015007, 9 pp. doi: 10.1088/0266-5611/27/1/015007.  Google Scholar

[15]

K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics, 28. Cambridge University Press, Cambridge, 1990. doi: 10.1017/CBO9780511526152.  Google Scholar

[16]

K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Monographs and Textbooks in Pure and Applied Mathematics, 83. Marcel Dekker, Inc., New York, 1984.  Google Scholar

[17]

O. Güler, On the convergence of the proximal point algorithm for convex minimization, SIAM J. Control Optim., 29 (1991), 403-419.  doi: 10.1137/0329022.  Google Scholar

[18]

J. M. Hendrickx and A. Olshevsky, Matrix $p$-Norms are NP-Hard to Approximate if $P\neq 1, 2, \infty$, SIAM. J. Matrix Anal. Appl., 31 (2010), 2802-2812.  doi: 10.1137/09076773X.  Google Scholar

[19]

S. Kamimura and W. Takahashi, Approximating solutions of maximal monotone operators in Hilbert spaces, J. Approx. Theory, 106 (2000), 226-240.  doi: 10.1006/jath.2000.3493.  Google Scholar

[20]

S. Kamimura and W. Takahashi, Strong convergence of a proximal-type algorithm in a Banach space, SIAM J. Optim., 13 (2002), 938-945.  doi: 10.1137/S105262340139611X.  Google Scholar

[21]

S. Kesornprom and P. Cholamjiak, Proximal type algorithms involving linesearch and inertial technique for split variational inclusion problem in Hilbert spaces with applications, Optimization, 68 (2019), 2365-2391.  doi: 10.1080/02331934.2019.1638389.  Google Scholar

[22]

G. L$\acute{o}$pez, V. Mart$\acute{i}$n-M$\acute{a}$rquez and H. K. Xu, Iterative algorithms for the multiple-sets split feasibility problem, Biomedical Mathematics: Promising Directions in Imaging, Therapy Planning and Inverse Problems, Medical Physics Publishing, Madison, (2010), 243–279. Google Scholar

[23]

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

[24]

B. Martinet, Régularisation d'inéquations variationnelles par approximations successives, Rev. Fr. Autom. Inform. Rech. Opér., 4 (1970), 154-158.   Google Scholar

[25]

E. Masad and S. Reich, A note on the multiple-set split convex feasibility problem in Hilbert space, J. Nonlinear Convex Anal., 8 (2008), 367-371.   Google Scholar

[26]

A. Moudafi, Split monotone variational inclusions, J. Optim. Theory Appl., 150 (2011), 275-283.  doi: 10.1007/s10957-011-9814-6.  Google Scholar

[27]

A. Moudafi and B. S. Thakur, Solving proximal split feasibility problems without prior knowledge of operator norms, Optim. Lett., 8 (2014), 2099-2110.  doi: 10.1007/s11590-013-0708-4.  Google Scholar

[28]

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

[29]

R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), 877-898.  doi: 10.1137/0314056.  Google Scholar

[30]

Y. Shehu and D. F. Agbebaku, On split inclusion problem and fixed point problem for multi-valued mappings, Comput. Appl. Math., 37 (2018), 1807-1824.  doi: 10.1007/s40314-017-0426-0.  Google Scholar

[31]

Y. Shehu and O. S. Iyiola, Accelerated hybrid viscosity and steepest-descent method for proximal split feasibility problems, Optimization, 67 (2018), 475-492.  doi: 10.1080/02331934.2017.1405955.  Google Scholar

[32]

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

[33]

Y. Shehu and O. S. Iyiola, Nonlinear iteration method for proximal split feasibility problems, Math. Methods Appl. Sci., 41 (2018), 781-802.  doi: 10.1002/mma.4644.  Google Scholar

[34]

Y. ShehuF. U. Ogbuisi and O. S. Iyiola, Convergence analysis of an iterative algorithm for fixed point problems and split feasibility problems in certain Banach spaces, Optimization, 65 (2016), 299-323.  doi: 10.1080/02331934.2015.1039533.  Google Scholar

[35]

M. V. Solodov and B. F. Svaiter, Forcing strong convergence of proximal point iterations in a Hilbert space, Math. Program, 87 (2000), 189-202.  doi: 10.1007/s101079900113.  Google Scholar

[36]

H. Stark, Image Recovery: Theory and Applications, Academic Press, Inc., Orlando, FL, 1987.  Google Scholar

[37]

W. Takahashi, Nonlinear Functional Analysis. Fixed Point Theory and Its Application, Yokohama Publishers, Yokohama, 2000.  Google Scholar

[38]

F. H. Wang and H.-K. Xu, Approximating curve and strong convergence of the $CQ$ algorithm for the split feasibility problem, J. Inequal. Appl., 2010 (2010), 102085, 13 pp. doi: 10.1155/2010/102085.  Google Scholar

[39]

H.-K. Xu, A variable Krasnosel$\acute{}$skii-Mann algorithm and the multiple-set split feasibility problem, Inverse Probl., 22 (2006), 2021-2034.  doi: 10.1088/0266-5611/22/6/007.  Google Scholar

[40]

H.-K. Xu, Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces, Inverse Probl., 26 (2010), 105018, 17 pp. doi: 10.1088/0266-5611/26/10/105018.  Google Scholar

[41]

Q. Z. Yang, The relaxed CQ algorithm for solving the split feasibility problem, Inverse Probl., 20 (2004), 1261-1266.  doi: 10.1088/0266-5611/20/4/014.  Google Scholar

[42]

L. Yang and F. H. Zhao, General split variational inclusion problem in Hilbert spaces, Abstr. Appl. Anal., (2014), 816035, 8 pp. doi: 10.1155/2014/816035.  Google Scholar

[43]

Y. H. YaoM. Postolache and Z. C. Zhu, Gradient methods with selection technique for the multiple-sets split feasibility problem, Optimization, 69 (2020), 269-281.  doi: 10.1080/02331934.2019.1602772.  Google Scholar

[44]

Y. H. YaoX. L. Qin and J.-C. Yao, Convergence analysis of an inertial iterate for the proximal split feasibility problem, J. Nonlinear Convex Anal., 20 (2019), 489-498.   Google Scholar

[45]

Y. H. YaoX. L. Qin and J.-C. Yao, Constructive approximation of solutions to proximal split feasibility problems, J. Nonlinear Convex Anal., 19 (2018), 2165-2175.   Google Scholar

[46]

Y. H. Yao, Z. S. Yao, A. A. N. Abdou and Y. J. Cho, Self-adaptive algorithms for proximal split feasibility problems and strong convergence analysis, Fixed Point Theory Appl., 2015 (2015), 13 pp. doi: 10.1186/s13663-015-0462-7.  Google Scholar

[47]

L. H. YenL. D. Muu and N. T. T. Huyen, An algorithm for a class of split feasibility problems: Application to a model in electricity production, Math. Meth. Oper. Res., 84 (2016), 549-565.  doi: 10.1007/s00186-016-0553-1.  Google Scholar

[48]

J. L. Zhao and Q. Z. Yang, Self-adaptive projection methods for the multiple-sets split feasibility problem, Inverse Probl., 27 (2011), 035009, 13 pp. doi: 10.1088/0266-5611/27/3/035009.  Google Scholar

show all references

References:
[1]

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

[2]

H. H. Bauschke and P. L. Combettes, A weak-to-strong convergence principle for Fej$\acute{e}$r monotone methods in Hilbert spaces, Math. Oper. Res., 26 (2001), 248-264.  doi: 10.1287/moor.26.2.248.10558.  Google Scholar

[3]

H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, New York, 2011. doi: 10.1007/978-1-4419-9467-7.  Google Scholar

[4]

H. Brézis and P.-L. Lions, Produits infinis de résolvantes, Israel J. Math., 29 (1978), 329-345.  doi: 10.1007/BF02761171.  Google Scholar

[5]

C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Probl., 18 (2002), 441-453.  doi: 10.1088/0266-5611/18/2/310.  Google Scholar

[6]

C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Probl., 20 (2004), 103-120.  doi: 10.1088/0266-5611/20/1/006.  Google Scholar

[7]

Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projection in a product space, Numer. Algorithms, 8 (1994), 221-239.  doi: 10.1007/BF02142692.  Google Scholar

[8]

Y. CensorT. BortfeldB. Martin and A. Trofimov, A unified approach for inversion problems in intensity modulated radiation therapy, Phys. Med. Biol., 51 (2003), 2353-2365.  doi: 10.1088/0031-9155/51/10/001.  Google Scholar

[9]

C.-S. Chuang, Strong convergence theorems for the split variational inclusion problem in Hilbert spaces, Fixed Point Theory and Appl., 2013 (2013), 20 pp. doi: 10.1186/1687-1812-2013-350.  Google Scholar

[10]

C.-S. Chuang, Simultaneous subgradient algorithms for the generalized split variational inclusion problem in Hilbert spaces, Numer. Funct. Anal. Optim., 38 (2017), 306-326.  doi: 10.1080/01630563.2016.1233120.  Google Scholar

[11]

C.-S. Chuang, Hybrid inertial proximal algorithm for the split variational inclusion problem in Hilbert spaces with applications, Optimization, 66 (2017), 777-792.  doi: 10.1080/02331934.2017.1306744.  Google Scholar

[12]

P. L. Combettes, The convex feasibility problem in image recovery, Advances in Imaging and Electron Physics, 95 (1996), 155-270.  doi: 10.1016/S1076-5670(08)70157-5.  Google Scholar

[13]

J. ContrerasM. Klusch and J. B. Krawczyk, Numerical solution to Nash-Cournot equilibria in coupled constraint electricity markets, IEEE Trans. Power Syst., 19 (2004), 195-206.  doi: 10.1109/TPWRS.2003.820692.  Google Scholar

[14]

Y. Z. Dang and Y. Gao, The strong convergence of a KM-CQ-like algorithm for a split feasibility problem, Inverse Probl., 27 (2011), 015007, 9 pp. doi: 10.1088/0266-5611/27/1/015007.  Google Scholar

[15]

K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics, 28. Cambridge University Press, Cambridge, 1990. doi: 10.1017/CBO9780511526152.  Google Scholar

[16]

K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Monographs and Textbooks in Pure and Applied Mathematics, 83. Marcel Dekker, Inc., New York, 1984.  Google Scholar

[17]

O. Güler, On the convergence of the proximal point algorithm for convex minimization, SIAM J. Control Optim., 29 (1991), 403-419.  doi: 10.1137/0329022.  Google Scholar

[18]

J. M. Hendrickx and A. Olshevsky, Matrix $p$-Norms are NP-Hard to Approximate if $P\neq 1, 2, \infty$, SIAM. J. Matrix Anal. Appl., 31 (2010), 2802-2812.  doi: 10.1137/09076773X.  Google Scholar

[19]

S. Kamimura and W. Takahashi, Approximating solutions of maximal monotone operators in Hilbert spaces, J. Approx. Theory, 106 (2000), 226-240.  doi: 10.1006/jath.2000.3493.  Google Scholar

[20]

S. Kamimura and W. Takahashi, Strong convergence of a proximal-type algorithm in a Banach space, SIAM J. Optim., 13 (2002), 938-945.  doi: 10.1137/S105262340139611X.  Google Scholar

[21]

S. Kesornprom and P. Cholamjiak, Proximal type algorithms involving linesearch and inertial technique for split variational inclusion problem in Hilbert spaces with applications, Optimization, 68 (2019), 2365-2391.  doi: 10.1080/02331934.2019.1638389.  Google Scholar

[22]

G. L$\acute{o}$pez, V. Mart$\acute{i}$n-M$\acute{a}$rquez and H. K. Xu, Iterative algorithms for the multiple-sets split feasibility problem, Biomedical Mathematics: Promising Directions in Imaging, Therapy Planning and Inverse Problems, Medical Physics Publishing, Madison, (2010), 243–279. Google Scholar

[23]

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

[24]

B. Martinet, Régularisation d'inéquations variationnelles par approximations successives, Rev. Fr. Autom. Inform. Rech. Opér., 4 (1970), 154-158.   Google Scholar

[25]

E. Masad and S. Reich, A note on the multiple-set split convex feasibility problem in Hilbert space, J. Nonlinear Convex Anal., 8 (2008), 367-371.   Google Scholar

[26]

A. Moudafi, Split monotone variational inclusions, J. Optim. Theory Appl., 150 (2011), 275-283.  doi: 10.1007/s10957-011-9814-6.  Google Scholar

[27]

A. Moudafi and B. S. Thakur, Solving proximal split feasibility problems without prior knowledge of operator norms, Optim. Lett., 8 (2014), 2099-2110.  doi: 10.1007/s11590-013-0708-4.  Google Scholar

[28]

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

[29]

R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), 877-898.  doi: 10.1137/0314056.  Google Scholar

[30]

Y. Shehu and D. F. Agbebaku, On split inclusion problem and fixed point problem for multi-valued mappings, Comput. Appl. Math., 37 (2018), 1807-1824.  doi: 10.1007/s40314-017-0426-0.  Google Scholar

[31]

Y. Shehu and O. S. Iyiola, Accelerated hybrid viscosity and steepest-descent method for proximal split feasibility problems, Optimization, 67 (2018), 475-492.  doi: 10.1080/02331934.2017.1405955.  Google Scholar

[32]

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

[33]

Y. Shehu and O. S. Iyiola, Nonlinear iteration method for proximal split feasibility problems, Math. Methods Appl. Sci., 41 (2018), 781-802.  doi: 10.1002/mma.4644.  Google Scholar

[34]

Y. ShehuF. U. Ogbuisi and O. S. Iyiola, Convergence analysis of an iterative algorithm for fixed point problems and split feasibility problems in certain Banach spaces, Optimization, 65 (2016), 299-323.  doi: 10.1080/02331934.2015.1039533.  Google Scholar

[35]

M. V. Solodov and B. F. Svaiter, Forcing strong convergence of proximal point iterations in a Hilbert space, Math. Program, 87 (2000), 189-202.  doi: 10.1007/s101079900113.  Google Scholar

[36]

H. Stark, Image Recovery: Theory and Applications, Academic Press, Inc., Orlando, FL, 1987.  Google Scholar

[37]

W. Takahashi, Nonlinear Functional Analysis. Fixed Point Theory and Its Application, Yokohama Publishers, Yokohama, 2000.  Google Scholar

[38]

F. H. Wang and H.-K. Xu, Approximating curve and strong convergence of the $CQ$ algorithm for the split feasibility problem, J. Inequal. Appl., 2010 (2010), 102085, 13 pp. doi: 10.1155/2010/102085.  Google Scholar

[39]

H.-K. Xu, A variable Krasnosel$\acute{}$skii-Mann algorithm and the multiple-set split feasibility problem, Inverse Probl., 22 (2006), 2021-2034.  doi: 10.1088/0266-5611/22/6/007.  Google Scholar

[40]

H.-K. Xu, Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces, Inverse Probl., 26 (2010), 105018, 17 pp. doi: 10.1088/0266-5611/26/10/105018.  Google Scholar

[41]

Q. Z. Yang, The relaxed CQ algorithm for solving the split feasibility problem, Inverse Probl., 20 (2004), 1261-1266.  doi: 10.1088/0266-5611/20/4/014.  Google Scholar

[42]

L. Yang and F. H. Zhao, General split variational inclusion problem in Hilbert spaces, Abstr. Appl. Anal., (2014), 816035, 8 pp. doi: 10.1155/2014/816035.  Google Scholar

[43]

Y. H. YaoM. Postolache and Z. C. Zhu, Gradient methods with selection technique for the multiple-sets split feasibility problem, Optimization, 69 (2020), 269-281.  doi: 10.1080/02331934.2019.1602772.  Google Scholar

[44]

Y. H. YaoX. L. Qin and J.-C. Yao, Convergence analysis of an inertial iterate for the proximal split feasibility problem, J. Nonlinear Convex Anal., 20 (2019), 489-498.   Google Scholar

[45]

Y. H. YaoX. L. Qin and J.-C. Yao, Constructive approximation of solutions to proximal split feasibility problems, J. Nonlinear Convex Anal., 19 (2018), 2165-2175.   Google Scholar

[46]

Y. H. Yao, Z. S. Yao, A. A. N. Abdou and Y. J. Cho, Self-adaptive algorithms for proximal split feasibility problems and strong convergence analysis, Fixed Point Theory Appl., 2015 (2015), 13 pp. doi: 10.1186/s13663-015-0462-7.  Google Scholar

[47]

L. H. YenL. D. Muu and N. T. T. Huyen, An algorithm for a class of split feasibility problems: Application to a model in electricity production, Math. Meth. Oper. Res., 84 (2016), 549-565.  doi: 10.1007/s00186-016-0553-1.  Google Scholar

[48]

J. L. Zhao and Q. Z. Yang, Self-adaptive projection methods for the multiple-sets split feasibility problem, Inverse Probl., 27 (2011), 035009, 13 pp. doi: 10.1088/0266-5611/27/3/035009.  Google Scholar

Table 1.  The results of Algorithm 3.1 with the difference of $\gamma_{n}$
NN.P. $\gamma_{n}$ Average iteration Average times
10 10 $\frac{n}{2(n+1)}$ 1385 36.2813
10 10 1 734 18.9375
10 10 2 387 9.5625
10 10 3 265 6.6094
10 10 $\frac{3(n+1)}{(n+2)}$ 202 5.0625
NN.P. $\gamma_{n}$ Average iteration Average times
10 10 $\frac{n}{2(n+1)}$ 1385 36.2813
10 10 1 734 18.9375
10 10 2 387 9.5625
10 10 3 265 6.6094
10 10 $\frac{3(n+1)}{(n+2)}$ 202 5.0625
Table 2.  The results computed on Algorithm 3.1 and the method in [47]
N N.P. Average iteration Average times
Algorithm 3.1 Method in [47] Algorithm 3.1 Method in [47]
10 10 202 921 5.0625 30.4219
20 10 114 14319 3.0781 403.1563
30 10 72 23114 2.6250 969.2813
50 10 43 26894 1.5625 1000.0190
100 10 17 45443 0.4844 3000.4453
N N.P. Average iteration Average times
Algorithm 3.1 Method in [47] Algorithm 3.1 Method in [47]
10 10 202 921 5.0625 30.4219
20 10 114 14319 3.0781 403.1563
30 10 72 23114 2.6250 969.2813
50 10 43 26894 1.5625 1000.0190
100 10 17 45443 0.4844 3000.4453
Table 3.  The results of Algorithm 3.1 with the difference of $\theta_{n}$
$\theta_{n}$ Average iteration Average times
0 431 4.9200
0.1 236 2.4900
0.25 223 2.300
0.5 80 0.6600
0.75 71 0.5538
0.9 54 0.4600
1 41 0.3700
$\theta_{n}$ Average iteration Average times
0 431 4.9200
0.1 236 2.4900
0.25 223 2.300
0.5 80 0.6600
0.75 71 0.5538
0.9 54 0.4600
1 41 0.3700
Table 4.  The results computed on Algorithm 3.1 and the method in [43]
N Average iteration Average times
Algorithm 3.1 Method in [43] Algorithm 3.1 Method in [43]
10 96 206 0.7400 1.9600
20 77 150 0.6100 1.3300
N Average iteration Average times
Algorithm 3.1 Method in [43] Algorithm 3.1 Method in [43]
10 96 206 0.7400 1.9600
20 77 150 0.6100 1.3300
Table 5.  The results computed in Algorithm 3.1 and the method in [44]
N Average iteration Average times
Algorithm 3.1 Method in [44] Algorithm 3.1 Method in [44]
10 38 150 0.7600 4.1100
20 10 129 0.2200 2.3600
N Average iteration Average times
Algorithm 3.1 Method in [44] Algorithm 3.1 Method in [44]
10 38 150 0.7600 4.1100
20 10 129 0.2200 2.3600
[1]

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