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  January 2019 Early access  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 and 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. 

[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.

show all references

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

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) $8.3807\times 10^{-2}$ $0.1546$ $0.20739$
[1]

Francis Akutsah, Akindele Adebayo Mebawondu, Hammed Anuoluwapo Abass, Ojen Kumar Narain. A self adaptive method for solving a class of bilevel variational inequalities with split variational inequality and composed fixed point problem constraints in Hilbert spaces. Numerical Algebra, Control and Optimization, 2021  doi: 10.3934/naco.2021046

[2]

Chibueze Christian Okeke, Abdulmalik Usman Bello, Lateef Olakunle Jolaoso, Kingsley Chimuanya Ukandu. Inertial method for split null point problems with pseudomonotone variational inequality problems. Numerical Algebra, Control and Optimization, 2021  doi: 10.3934/naco.2021037

[3]

Shaotao Hu, Yuanheng Wang, Bing Tan, Fenghui Wang. Inertial iterative method for solving variational inequality problems of pseudo-monotone operators and fixed point problems of nonexpansive mappings in Hilbert spaces. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022060

[4]

Habib ur Rehman, Poom Kumam, Yusuf I. Suleiman, Widaya Kumam. An adaptive block iterative process for a class of multiple sets split variational inequality problems and common fixed point problems in Hilbert spaces. Numerical Algebra, Control and Optimization, 2022  doi: 10.3934/naco.2022007

[5]

G. M. de Araújo, S. B. de Menezes. On a variational inequality for the Navier-Stokes operator with variable viscosity. Communications on Pure and Applied Analysis, 2006, 5 (3) : 583-596. doi: 10.3934/cpaa.2006.5.583

[6]

Sylvia Serfaty. Gamma-convergence of gradient flows on Hilbert and metric spaces and applications. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1427-1451. doi: 10.3934/dcds.2011.31.1427

[7]

S. J. Li, Z. M. Fang. On the stability of a dual weak vector variational inequality problem. Journal of Industrial and Management Optimization, 2008, 4 (1) : 155-165. doi: 10.3934/jimo.2008.4.155

[8]

Ren-You Zhong, Nan-Jing Huang. Strict feasibility for generalized mixed variational inequality in reflexive Banach spaces. Numerical Algebra, Control and Optimization, 2011, 1 (2) : 261-274. doi: 10.3934/naco.2011.1.261

[9]

Anne-Laure Bessoud. A variational convergence for bifunctionals. Application to a model of strong junction. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 399-417. doi: 10.3934/dcdss.2012.5.399

[10]

Zhili Ge, Gang Qian, Deren Han. Global convergence of an inexact operator splitting method for monotone variational inequalities. Journal of Industrial and Management Optimization, 2011, 7 (4) : 1013-1026. doi: 10.3934/jimo.2011.7.1013

[11]

Pengyu Chen, Yongxiang Li, Xuping Zhang. On the initial value problem of fractional stochastic evolution equations in Hilbert spaces. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1817-1840. doi: 10.3934/cpaa.2015.14.1817

[12]

Matthias Gerdts, Stefan Horn, Sven-Joachim Kimmerle. Line search globalization of a semismooth Newton method for operator equations in Hilbert spaces with applications in optimal control. Journal of Industrial and Management Optimization, 2017, 13 (1) : 47-62. doi: 10.3934/jimo.2016003

[13]

T. A. Shaposhnikova, M. N. Zubova. Homogenization problem for a parabolic variational inequality with constraints on subsets situated on the boundary of the domain. Networks and Heterogeneous Media, 2008, 3 (3) : 675-689. doi: 10.3934/nhm.2008.3.675

[14]

Junfeng Yang. Dynamic power price problem: An inverse variational inequality approach. Journal of Industrial and Management Optimization, 2008, 4 (4) : 673-684. doi: 10.3934/jimo.2008.4.673

[15]

Jianlin Jiang, Shun Zhang, Su Zhang, Jie Wen. A variational inequality approach for constrained multifacility Weber problem under gauge. Journal of Industrial and Management Optimization, 2018, 14 (3) : 1085-1104. doi: 10.3934/jimo.2017091

[16]

Yarui Duan, Pengcheng Wu, Yuying Zhou. Penalty approximation method for a double obstacle quasilinear parabolic variational inequality problem. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022017

[17]

Ouafa Belguidoum, Hassina Grar. An improved projection algorithm for variational inequality problem with multivalued mapping. Numerical Algebra, Control and Optimization, 2022  doi: 10.3934/naco.2022002

[18]

Irene Benedetti, Luisa Malaguti, Valentina Taddei. Nonlocal problems in Hilbert spaces. Conference Publications, 2015, 2015 (special) : 103-111. doi: 10.3934/proc.2015.0103

[19]

Fritz Gesztesy, Rudi Weikard, Maxim Zinchenko. On a class of model Hilbert spaces. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5067-5088. doi: 10.3934/dcds.2013.33.5067

[20]

Liping Pang, Fanyun Meng, Jinhe Wang. Asymptotic convergence of stationary points of stochastic multiobjective programs with parametric variational inequality constraint via SAA approach. Journal of Industrial and Management Optimization, 2019, 15 (4) : 1653-1675. doi: 10.3934/jimo.2018116

2021 Impact Factor: 1.411

Metrics

  • PDF downloads (546)
  • HTML views (1889)
  • Cited by (3)

Other articles
by authors

[Back to Top]