January  2019, 15(1): 319-342. 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, 2019, 15 (1) : 319-342. 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.   Google Scholar

[2]

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

[3]

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

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[28]

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

[29]

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

[30]

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

[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.  Google Scholar

[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.  Google Scholar

[33]

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

[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.  Google Scholar

[35]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[43]

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

[44]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[48]

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

[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.  Google Scholar

[50]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.   Google Scholar

[2]

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

[3]

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

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[28]

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

[29]

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

[30]

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

[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.  Google Scholar

[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.  Google Scholar

[33]

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

[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.  Google Scholar

[35]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[43]

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

[44]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[48]

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

[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.  Google Scholar

[50]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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