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doi: 10.3934/jimo.2018045

On a modified extragradient method for variational inequality problem with application to industrial electricity production

1. 

University of Nigeria, Department of Mathematics, Nsukka, Nigeria

2. 

Institute of Mathematics, University of Würzburg, Campus Hubland Nord, Emil-Fischer-Str. 30, 97074 Würzburg, Germany

3. 

Department of Mathematics, Minnesota State University, Moorhead, Minnesota, USA

* Corresponding author: Yekini Shehu

The first author is supported by the Alexander von Humboldt-Foundation.

Received  June 2017 Revised  October 2017 Published  April 2018

In this paper, we present a modified extragradient-type method for solving the variational inequality problem involving uniformly continuous pseudomonotone operator. It is shown that under certain mild assumptions, this method is strongly convergent in infinite dimensional real Hilbert spaces. We give some numerical computational experiments which involve a comparison of our proposed method with other existing method in a model on industrial electricity production.

Citation: Yekini Shehu, Olaniyi Iyiola. On a modified extragradient method for variational inequality problem with application to industrial electricity production. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2018045
References:
[1]

R. Y. ApostolA. A. Grynenko and V. V. Semenov, Iterative algorithms for monotone bilevel variational inequalities, J. Comp. Appl. Math., 107 (2012), 3-14.

[2]

K. J. Arrow and G. Debreu, Existence of an equilibrium for a competitive economy, Econometrica, 22 (1954), 265-290. doi: 10.2307/1907353.

[3]

J. -P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Wiley, New York, 1984.

[4]

C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities; Applications to Free Boundary Problems, Wiley, New York, 1984.

[5]

H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, CMS Books in Mathematics, Springer, New York, 2011.

[6]

J. Y. Bello Cruz and A. N. Iusem, A strongly convergent direct method for monotone variational inequalities in Hilbert spaces, Num. Funct. Anal. Optim., 30 (2009), 23-36. doi: 10.1080/01630560902735223.

[7]

J. Y. Bello Cruz and A. N. Iusem, An explicit algorithm for monotone variational inequalities, Optim., 61 (2012), 855-871. doi: 10.1080/02331934.2010.536232.

[8]

F. Browder, Nonlinear monotone operators and convex sets in Banach spaces, Bull. Amer. Math. Soc., 71 (1965), 780-785. doi: 10.1090/S0002-9904-1965-11391-X.

[9]

X. CaiG. Gu and B. He, On the O(1/t) convergence rate of the projection and contraction methods for variational inequalities with Lipschitz continuous monotone operators, Comput. Optim. Appl., 57 (2014), 339-363. doi: 10.1007/s10589-013-9599-7.

[10]

A. Cegielski, Iterative Methods for Fixed Point Problems in Hilbert Spaces, Lecture Notes in Mathematics 2057, Springer, Berlin, 2012.

[11]

L. C. CengN. Hadjisavvas and N.-C. Wong, Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems, J. Glob. Optim., 46 (2010), 635-646. doi: 10.1007/s10898-009-9454-7.

[12]

L. C. CengM. Teboulle and J. C. Yao, Weak convergence of an iterative method for pseudomonotone variational inequalities and fixed-point problems, J. Optim. Theory Appl., 146 (2010), 19-31. doi: 10.1007/s10957-010-9650-0.

[13]

Y. CensorA. Gibali and S. Reich, Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space, Optim. Methods Softw., 26 (2011), 827-845. doi: 10.1080/10556788.2010.551536.

[14]

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

[15]

F. FacchineiA. Fischer and V. Piccialli, On generalized Nash games and variational inequalities, Oper. Res. Lett., 35 (2007), 159-164. doi: 10.1016/j.orl.2006.03.004.

[16]

F. Facchinei and J. -S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Volume Ⅰ. Springer Series in Operations Research, Springer, New York, 2003.

[17]

R. Glowinski, J. -L. Lions and R. Trémolières, Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, 1981.

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P. T. Harker and J. -S. Pang, A damped-Newton method for the linear complementarity problem, in Computational Solution of Nonlinear Systems of Equations, Lectures in Appl. Math., G. Allgower and K. Georg, eds., AMS, Providence, RI, 26 (1990), 265-284.

[19]

P. T. Harker, Generalized Nash games and quasi-variational inequalities, European J. Oper. Res., 54 (1991), 81-94. doi: 10.1016/0377-2217(91)90325-P.

[20]

Ph. Hartman and G. Stampacchia, On some non linear elliptic differential functional equations, Acta Math., 115 (1966), 271-310. doi: 10.1007/BF02392210.

[21]

B.-S. HeZ.-H. Yang and X.-M. Yuan, An approximate proximal-extragradient type method for monotone variational inequalities, J. Math. Anal. Appl., 300 (2004), 362-374. doi: 10.1016/j.jmaa.2004.04.068.

[22]

D. V. HieuP. K. Anh and L. D. Muu, Modified hybrid projection methods for finding common solutions to variational inequality problems, Comput. Optim. Appl., 66 (2017), 75-96. doi: 10.1007/s10589-016-9857-6.

[23]

B. F. Hobbs, Linear complementarity models of Nash-Cournot competition in bilateral and POOLCO power market, IEEE Trans. Power Syst., 16 (2001), 194-202.

[24]

A. N. 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.

[25]

A. N. Iusem and M. Nasri, Korpelevich's method for variational inequality problems in Banach spaces, J. Glob. Optim., 50 (2011), 59-76. doi: 10.1007/s10898-010-9613-x.

[26]

A. N. Iusem and B. F. Svaiter, A variant of Korpelevich's method for variational inequalities with a new search strategy, Optimization, 42 (1997), 309-321. doi: 10.1080/02331939708844365.

[27]

E. N. Khobotov, A modification of the extragradient method for solving variational inequalities and certain optimization problems, Zh. Vychisl. Mat. i Mat. Fiz., 27 (1989), 1462-1473,1597.

[28]

D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980.

[29]

I. V. Konnov, Combined Relaxation Methods for Variational Inequalities, Springer-Verlag, Berlin, 2001.

[30]

G. M. Korpelevich, The extragradient method for finding saddle points and other problems, Ekonomika i Mat. Metody, 12 (1976), 747-756.

[31]

R. Kraikaew and S. Saejung, Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Hilbert spaces, J. Optim. Theory Appl., 163 (2014), 399-412. doi: 10.1007/s10957-013-0494-2.

[32]

P.-E. Maingé and M. L. Gobinddass, Convergence of one-step projected gradient methods for variational inequalities, J. Optim. Theory Appl., 171 (2016), 146-168. doi: 10.1007/s10957-016-0972-4.

[33]

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

[34]

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

[35]

P. Marcotte, Applications of Khobotov's algorithm to variational and network equlibrium problems, Inf. Syst. Oper. Res., 29 (1991), 258-270.

[36]

J. Mashreghi and M. Nasri, Forcing strong convergence of Korpelevich's method in Banach spaces with its applications in game theory, Nonlinear Analysis, 72 (2010), 2086-2099. doi: 10.1016/j.na.2009.10.009.

[37]

N. Nadezhkina and W. Takahashi, Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings, SIAM J. Optim., 16 (2006), 1230-1241. doi: 10.1137/050624315.

[38]

N. Nadezhkina and W. Takahashi, Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 128 (2006), 191-201. doi: 10.1007/s10957-005-7564-z.

[39]

S. M. Robinson, Shadow prices for measures of effectiveness, Ⅰ: Linear model, Oper. Res., 41 (1993), 518-535. doi: 10.1287/opre.41.3.518.

[40]

S. M. Robinson, Shadow prices for measures of effectiveness, Ⅱ: General model, Oper. Res., 41 (1993), 536-548. doi: 10.1287/opre.41.3.536.

[41]

Y. Shehu and O. S. Iyiola, Strong convergence result for monotone variational inequalities, Numer. Algor., 76 (2017), 259-282. doi: 10.1007/s11075-016-0253-1.

[42]

M. V. Solodov and B. F. Svaiter, A new projection method for variational inequality problems, SIAM J. Control Optim., 37 (1999), 765-776. doi: 10.1137/S0363012997317475.

[43]

D. Sun, An iterative method for solving variational inquality problems and complementarity problems, Numer. Math. J. Chinese Univ., 16 (1994), 145-153.

[44]

W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, 2000.

[45]

P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control Optim., 38 (2000), 431-446. doi: 10.1137/S0363012998338806.

[46]

Y. J. WangN. H. Xiu and C. Y. Wang, Unified framework for extragradient-type methods for pseudomonotone variational inequalities, J. Optim. Theory Appl., 111 (2001), 641-656. doi: 10.1023/A:1012606212823.

[47]

Y. J. WangN. H. Xiu and C. Y. Wang, A new version of extragradient method for variational inequality problems, Comput. Math. Appl., 42 (2001), 969-979. doi: 10.1016/S0898-1221(01)00213-9.

[48]

J.-Y. Wei and Y. Smeers, Spatial oligopolistic electricity models with Cournot generators and regulated transmission prices, Oper. Res., 47 (1999), 102-112.

[49]

G. L. Xue and Y. Y. Ye, An efficient algorithm for minimizing a sum of Euclidean norms with applications, SIAM J. Optim., 7 (1997), 1017-1036. doi: 10.1137/S1052623495288362.

[50]

H.-K. Xu, Iterative algorithms for nonlinear operators, J. London. Math. Soc., 66 (2002), 240-256. doi: 10.1112/S0024610702003332.

[51]

Y. YaoG. Marino and L. Muglia, A modified Korpelevich's method convergent to the minimum-norm solution of a variational inequality, Optimization, 63 (2014), 559-569. doi: 10.1080/02331934.2012.674947.

[52]

Y. Yao and M. Postolache, Iterative methods for pseudomonotone variational inequalities and fixed-point problems, J. Optim. Theory Appl., 155 (2012), 273-287. doi: 10.1007/s10957-012-0055-0.

[53]

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.

[54]

J. ZhangB. Qu and N. Xiu, Some projection-like methods for the generalized Nash equilibria, Comput. Optim. Appl., 45 (2010), 89-109. doi: 10.1007/s10589-008-9173-x.

show all references

References:
[1]

R. Y. ApostolA. A. Grynenko and V. V. Semenov, Iterative algorithms for monotone bilevel variational inequalities, J. Comp. Appl. Math., 107 (2012), 3-14.

[2]

K. J. Arrow and G. Debreu, Existence of an equilibrium for a competitive economy, Econometrica, 22 (1954), 265-290. doi: 10.2307/1907353.

[3]

J. -P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Wiley, New York, 1984.

[4]

C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities; Applications to Free Boundary Problems, Wiley, New York, 1984.

[5]

H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, CMS Books in Mathematics, Springer, New York, 2011.

[6]

J. Y. Bello Cruz and A. N. Iusem, A strongly convergent direct method for monotone variational inequalities in Hilbert spaces, Num. Funct. Anal. Optim., 30 (2009), 23-36. doi: 10.1080/01630560902735223.

[7]

J. Y. Bello Cruz and A. N. Iusem, An explicit algorithm for monotone variational inequalities, Optim., 61 (2012), 855-871. doi: 10.1080/02331934.2010.536232.

[8]

F. Browder, Nonlinear monotone operators and convex sets in Banach spaces, Bull. Amer. Math. Soc., 71 (1965), 780-785. doi: 10.1090/S0002-9904-1965-11391-X.

[9]

X. CaiG. Gu and B. He, On the O(1/t) convergence rate of the projection and contraction methods for variational inequalities with Lipschitz continuous monotone operators, Comput. Optim. Appl., 57 (2014), 339-363. doi: 10.1007/s10589-013-9599-7.

[10]

A. Cegielski, Iterative Methods for Fixed Point Problems in Hilbert Spaces, Lecture Notes in Mathematics 2057, Springer, Berlin, 2012.

[11]

L. C. CengN. Hadjisavvas and N.-C. Wong, Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems, J. Glob. Optim., 46 (2010), 635-646. doi: 10.1007/s10898-009-9454-7.

[12]

L. C. CengM. Teboulle and J. C. Yao, Weak convergence of an iterative method for pseudomonotone variational inequalities and fixed-point problems, J. Optim. Theory Appl., 146 (2010), 19-31. doi: 10.1007/s10957-010-9650-0.

[13]

Y. CensorA. Gibali and S. Reich, Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space, Optim. Methods Softw., 26 (2011), 827-845. doi: 10.1080/10556788.2010.551536.

[14]

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

[15]

F. FacchineiA. Fischer and V. Piccialli, On generalized Nash games and variational inequalities, Oper. Res. Lett., 35 (2007), 159-164. doi: 10.1016/j.orl.2006.03.004.

[16]

F. Facchinei and J. -S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Volume Ⅰ. Springer Series in Operations Research, Springer, New York, 2003.

[17]

R. Glowinski, J. -L. Lions and R. Trémolières, Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, 1981.

[18]

P. T. Harker and J. -S. Pang, A damped-Newton method for the linear complementarity problem, in Computational Solution of Nonlinear Systems of Equations, Lectures in Appl. Math., G. Allgower and K. Georg, eds., AMS, Providence, RI, 26 (1990), 265-284.

[19]

P. T. Harker, Generalized Nash games and quasi-variational inequalities, European J. Oper. Res., 54 (1991), 81-94. doi: 10.1016/0377-2217(91)90325-P.

[20]

Ph. Hartman and G. Stampacchia, On some non linear elliptic differential functional equations, Acta Math., 115 (1966), 271-310. doi: 10.1007/BF02392210.

[21]

B.-S. HeZ.-H. Yang and X.-M. Yuan, An approximate proximal-extragradient type method for monotone variational inequalities, J. Math. Anal. Appl., 300 (2004), 362-374. doi: 10.1016/j.jmaa.2004.04.068.

[22]

D. V. HieuP. K. Anh and L. D. Muu, Modified hybrid projection methods for finding common solutions to variational inequality problems, Comput. Optim. Appl., 66 (2017), 75-96. doi: 10.1007/s10589-016-9857-6.

[23]

B. F. Hobbs, Linear complementarity models of Nash-Cournot competition in bilateral and POOLCO power market, IEEE Trans. Power Syst., 16 (2001), 194-202.

[24]

A. N. 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.

[25]

A. N. Iusem and M. Nasri, Korpelevich's method for variational inequality problems in Banach spaces, J. Glob. Optim., 50 (2011), 59-76. doi: 10.1007/s10898-010-9613-x.

[26]

A. N. Iusem and B. F. Svaiter, A variant of Korpelevich's method for variational inequalities with a new search strategy, Optimization, 42 (1997), 309-321. doi: 10.1080/02331939708844365.

[27]

E. N. Khobotov, A modification of the extragradient method for solving variational inequalities and certain optimization problems, Zh. Vychisl. Mat. i Mat. Fiz., 27 (1989), 1462-1473,1597.

[28]

D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980.

[29]

I. V. Konnov, Combined Relaxation Methods for Variational Inequalities, Springer-Verlag, Berlin, 2001.

[30]

G. M. Korpelevich, The extragradient method for finding saddle points and other problems, Ekonomika i Mat. Metody, 12 (1976), 747-756.

[31]

R. Kraikaew and S. Saejung, Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Hilbert spaces, J. Optim. Theory Appl., 163 (2014), 399-412. doi: 10.1007/s10957-013-0494-2.

[32]

P.-E. Maingé and M. L. Gobinddass, Convergence of one-step projected gradient methods for variational inequalities, J. Optim. Theory Appl., 171 (2016), 146-168. doi: 10.1007/s10957-016-0972-4.

[33]

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

[34]

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

[35]

P. Marcotte, Applications of Khobotov's algorithm to variational and network equlibrium problems, Inf. Syst. Oper. Res., 29 (1991), 258-270.

[36]

J. Mashreghi and M. Nasri, Forcing strong convergence of Korpelevich's method in Banach spaces with its applications in game theory, Nonlinear Analysis, 72 (2010), 2086-2099. doi: 10.1016/j.na.2009.10.009.

[37]

N. Nadezhkina and W. Takahashi, Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings, SIAM J. Optim., 16 (2006), 1230-1241. doi: 10.1137/050624315.

[38]

N. Nadezhkina and W. Takahashi, Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 128 (2006), 191-201. doi: 10.1007/s10957-005-7564-z.

[39]

S. M. Robinson, Shadow prices for measures of effectiveness, Ⅰ: Linear model, Oper. Res., 41 (1993), 518-535. doi: 10.1287/opre.41.3.518.

[40]

S. M. Robinson, Shadow prices for measures of effectiveness, Ⅱ: General model, Oper. Res., 41 (1993), 536-548. doi: 10.1287/opre.41.3.536.

[41]

Y. Shehu and O. S. Iyiola, Strong convergence result for monotone variational inequalities, Numer. Algor., 76 (2017), 259-282. doi: 10.1007/s11075-016-0253-1.

[42]

M. V. Solodov and B. F. Svaiter, A new projection method for variational inequality problems, SIAM J. Control Optim., 37 (1999), 765-776. doi: 10.1137/S0363012997317475.

[43]

D. Sun, An iterative method for solving variational inquality problems and complementarity problems, Numer. Math. J. Chinese Univ., 16 (1994), 145-153.

[44]

W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, 2000.

[45]

P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control Optim., 38 (2000), 431-446. doi: 10.1137/S0363012998338806.

[46]

Y. J. WangN. H. Xiu and C. Y. Wang, Unified framework for extragradient-type methods for pseudomonotone variational inequalities, J. Optim. Theory Appl., 111 (2001), 641-656. doi: 10.1023/A:1012606212823.

[47]

Y. J. WangN. H. Xiu and C. Y. Wang, A new version of extragradient method for variational inequality problems, Comput. Math. Appl., 42 (2001), 969-979. doi: 10.1016/S0898-1221(01)00213-9.

[48]

J.-Y. Wei and Y. Smeers, Spatial oligopolistic electricity models with Cournot generators and regulated transmission prices, Oper. Res., 47 (1999), 102-112.

[49]

G. L. Xue and Y. Y. Ye, An efficient algorithm for minimizing a sum of Euclidean norms with applications, SIAM J. Optim., 7 (1997), 1017-1036. doi: 10.1137/S1052623495288362.

[50]

H.-K. Xu, Iterative algorithms for nonlinear operators, J. London. Math. Soc., 66 (2002), 240-256. doi: 10.1112/S0024610702003332.

[51]

Y. YaoG. Marino and L. Muglia, A modified Korpelevich's method convergent to the minimum-norm solution of a variational inequality, Optimization, 63 (2014), 559-569. doi: 10.1080/02331934.2012.674947.

[52]

Y. Yao and M. Postolache, Iterative methods for pseudomonotone variational inequalities and fixed-point problems, J. Optim. Theory Appl., 155 (2012), 273-287. doi: 10.1007/s10957-012-0055-0.

[53]

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.

[54]

J. ZhangB. Qu and N. Xiu, Some projection-like methods for the generalized Nash equilibria, Comput. Optim. Appl., 45 (2010), 89-109. doi: 10.1007/s10589-008-9173-x.

Figure 3.  Algorithm (5.2) with $\rho=0.3$
Figure 6.  Algorithm (5.2) with $\rho=0.8$
Figure 9.  Algorithm (5.2) with $\rho=1.2$
Figure 12.  Algorithm (5.2) with $\rho=1.6$
Figure 13.  Algorithm (5.3) Case Ⅰ
Figure 14.  Algorithm (5.3) Case Ⅱ
Figure 15.  Algorithm (5.3) Case Ⅲ
Table 2.  Algorithm (5.2) with different values of $\rho$
No. of Iterations CPU (Time)
$\rho = 0.3$ 5 0.0163
$\rho = 0.8$ 10 0.0372
$\rho = 1.2$ 9 0.0309
$\rho = 1.6$ 8 0.0158
No. of Iterations CPU (Time)
$\rho = 0.3$ 5 0.0163
$\rho = 0.8$ 10 0.0372
$\rho = 1.2$ 9 0.0309
$\rho = 1.6$ 8 0.0158
Table 3.  Algorithm (5.3) with different Cases
No. of Iterations CPU Time
Case Ⅰ 14 0.0045
Case Ⅱ 14 0.0043
Case Ⅲ 14 0.0049
No. of Iterations CPU Time
Case Ⅰ 14 0.0045
Case Ⅱ 14 0.0043
Case Ⅲ 14 0.0049
Table 1.  Comparison of our proposed algorithm with YNE algorithm (5.1) for different values of $N$
$N$ 4 10 20
Our Proposed Alg. 3.1 No. of Iter. 2 2 2
cpu (Time) $1.0652\times 10^{-3}$ $9.0633\times 10^{-4}$ $1.2178\times 10^{-3}$
YNE Alg. No. of Iter. 150 138 133
cpu (Time) $8.3807\times 10^{-2}$ $0.1546$ $0.20739$
$N$ 4 10 20
Our Proposed Alg. 3.1 No. of Iter. 2 2 2
cpu (Time) $1.0652\times 10^{-3}$ $9.0633\times 10^{-4}$ $1.2178\times 10^{-3}$
YNE Alg. No. of Iter. 150 138 133
cpu (Time) $8.3807\times 10^{-2}$ $0.1546$ $0.20739$
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