# American Institute of Mathematical Sciences

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:

show all references

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

##### References:
Algorithm (5.2) with $\rho=0.3$
Algorithm (5.2) with $\rho=0.8$
Algorithm (5.2) with $\rho=1.2$
Algorithm (5.2) with $\rho=1.6$
Algorithm (5.3) Case Ⅰ
Algorithm (5.3) Case Ⅱ
Algorithm (5.3) Case Ⅲ
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
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
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