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November  2021, 17(6): 3357-3371. doi: 10.3934/jimo.2020123

New inertial method for generalized split variational inclusion problems

Preeyanuch Chuasuk 1, , Ferdinard Ogbuisi 2,3,, , Yekini Shehu 4, and  Prasit Cholamjiak 5,

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  November 2021 Early access  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.

Keywords: Maximal monotone operator, resolvent operator, inertial method, weak convergence, Hilbert space.
Mathematics Subject Classification: Primary: 49J53, 65K10; Secondary: 49M37, 90C25.
Citation: Preeyanuch Chuasuk, Ferdinard Ogbuisi, Yekini Shehu, Prasit Cholamjiak. New inertial method for generalized split variational inclusion problems. Journal of Industrial and Management Optimization, 2021, 17 (6) : 3357-3371. 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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[8]

Y. Censor, T. Bortfeld, B. 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.

[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.

[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.

[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.

[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.

[13]

J. Contreras, M. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

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

[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. 

[26]

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

[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.

[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.

[29]

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

[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.

[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.

[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.

[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.

[34]

Y. Shehu, F. 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.

[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.

[36]

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

[37]

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

[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.

[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.

[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.

[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.

[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.

[43]

Y. H. Yao, M. 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.

[44]

Y. H. Yao, X. 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. 

[45]

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

[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.

[47]

L. H. Yen, L. 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.

[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.

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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[8]

Y. Censor, T. Bortfeld, B. 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.

[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.

[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.

[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.

[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.

[13]

J. Contreras, M. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

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

[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. 

[26]

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

[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.

[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.

[29]

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

[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.

[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.

[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.

[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.

[34]

Y. Shehu, F. 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.

[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.

[36]

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

[37]

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

[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.

[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.

[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.

[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.

[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.

[43]

Y. H. Yao, M. 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.

[44]

Y. H. Yao, X. 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. 

[45]

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

[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.

[47]

L. H. Yen, L. 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.

[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.

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
Table Options

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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
Table Options

Download as excel

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
Table Options

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$\theta_{n}$ 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
Table Options

Download as excel

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
Table Options

Download as excel

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
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