# 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  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:

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##### 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) $8.3807\times 10^{-2}$ $0.1546$ $0.20739$
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