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doi: 10.3934/jimo.2021095
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Inertial Tseng's extragradient method for solving variational inequality problems of pseudo-monotone and non-Lipschitz operators

 1 School of Mathematics Science, Chongqing Normal University, Chongqing 401331, China 2 Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China 3 Department of Mathematics and Physical Sciences, , California University of Pennsylvania, PA, USA

* Corresponding author: G. Cai

Received  September 2020 Revised  January 2021 Early access May 2021

Fund Project: The first author is supported by the NSF of China (Grant No. 11771063), the Natural Science Foundation of Chongqing(cstc2020jcyj-msxmX0455), Science and Technology Project of Chongqing Education Committee (Grant No. KJZD-K201900504) and the Program of Chongqing Innovation Research Group Project in University (Grant no. CXQT19018)

In this paper, we propose a new inertial Tseng's extragradient iterative algorithm for solving variational inequality problems of pseudo-monotone and non-Lipschitz operator in real Hilbert spaces. We prove that the sequence generated by proposed algorithm converges strongly to an element of solutions of variational inequality problem under some suitable assumptions imposed on the parameters. Finally, we give some numerical experiments for supporting our main results. The main results obtained in this paper extend and improve some related works in the literature.

Citation: Gang Cai, Yekini Shehu, Olaniyi S. Iyiola. Inertial Tseng's extragradient method for solving variational inequality problems of pseudo-monotone and non-Lipschitz operators. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021095
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References:
Example 1: $k = 20$, $N = 10$
Example 1: $k = 20$, $N = 20$
Example 1: $k = 20$, $N = 30$
Example 1: $k = 20$, $N = 40$
Example 1: $k = 30$, $N = 10$
Example 1: $k = 30$, $N = 20$
Example 1: $k = 30$, $N = 30$
Example 1: $k = 30$, $N = 40$
Example 1: Different $\gamma$ with $(N, k) = (20, 10)$
Example 1: Different $\gamma$ with $(N, k) = (20, 21)$
Example 1: Different $\gamma$ with $(N, k) = (20, 30)$
Example 1: Different $\gamma$ with $(N, k) = (20, 40)$
Example 1: Different $\gamma$ with $(N, k) = (30, 10)$
Example 1: Different $\gamma$ with $(N, k) = (30, 20)$
Example 1: Different $\gamma$ with $(N, k) = (30, 30)$
Example 1: Different $\gamma$ with $(N, k) = (30, 40)$
Example 1: Different $\mu$ with $(N, k) = (20, 10)$
Example 1: Different $\mu$ with $(N, k) = (20, 20)$
Example 1: Different $\mu$ with $(N, k) = (20, 30)$
Example 1: Different $\mu$ with $(N, k) = (20, 40)$
Example 1: Different $\mu$ with $(N, k) = (30, 10)$
Example 1: Different $\mu$ with $(N, k) = (30, 20)$
Example 1: Different $\mu$ with $(N, k) = (30, 30)$
Example 1: Different $\mu$ with $(N, k) = (30, 40)$
Example 2: Case I
Example 2: Case II
Example 2: Case III
Example 2: Case IV
Example 2: Case V
Example 2: Case VI
Example 2: Case I with different $\gamma$
Example 2: Case II with different $\gamma$
Example 2: Case III with different $\gamma$
Example 2: Case IV with different $\gamma$
Example 2: Case V with different $\gamma$
Example 2: Case VI with different $\gamma$
Example 2: Case I with different $\mu$
Example 2: Case II with different $\mu$
Example 2: Case III with different $\mu$
Example 2: Case IV with different $\mu$
Example 2: Case V with different $\mu$
Example 2: Case VI with different $\mu$
The value of error versus the iteration numbers for Example 3
Methods Parameters Choice for Comparison
 Proposed Alg. $\epsilon_n = \frac{1}{n^2}$ $\theta = 0.1$ $l=0.001$ $\beta_n = \frac{1}{n}$ $\gamma = 0.99$ $\mu = 0.99$ Thong Alg. (1) $\epsilon_n = \frac{1}{(n + 1)^2}$ $\theta = 0.1$ $\beta_n = \frac{1}{n + 1}$ $\lambda = \frac{1}{1.01L}$ Thong Alg. (2) $l=0.001$ $\gamma = 0.99$ $\mu = 0.99$ Thong Alg. (3) $\alpha_n = \frac{1}{n + 1}$ $l=0.001$ $\gamma = 0.99$ $\mu = 0.99$ Gibali Alg. $\alpha_n = \frac{1}{n + 1}$ $l=0.001$ $\gamma = 0.99$ $\mu = 0.99$
 Proposed Alg. $\epsilon_n = \frac{1}{n^2}$ $\theta = 0.1$ $l=0.001$ $\beta_n = \frac{1}{n}$ $\gamma = 0.99$ $\mu = 0.99$ Thong Alg. (1) $\epsilon_n = \frac{1}{(n + 1)^2}$ $\theta = 0.1$ $\beta_n = \frac{1}{n + 1}$ $\lambda = \frac{1}{1.01L}$ Thong Alg. (2) $l=0.001$ $\gamma = 0.99$ $\mu = 0.99$ Thong Alg. (3) $\alpha_n = \frac{1}{n + 1}$ $l=0.001$ $\gamma = 0.99$ $\mu = 0.99$ Gibali Alg. $\alpha_n = \frac{1}{n + 1}$ $l=0.001$ $\gamma = 0.99$ $\mu = 0.99$
Example 1: Comparison among methods with different values of $N$ and $k$
 $N=10$ $N=20$ $N=30$ $N=40$ $k=20$ Iter. Time Iter. Time Iter. Time Iter. Time Proposed Alg. 3 1.3843 3 1.7672 3 1.7564 4 2.2017 Thong Alg. (1) 76 1.2902 139 2.7111 111 2.1715 232 37.7743 Thong Alg. (2) 2136 36.6812 1561 30.7776 1370 31.8672 1160 4.0453 Thong Alg. (3) 86 1.1655 152 2.3615 148 2.4878 178 29.0789 Gibali Alg. 150 12.0085 235 20.3243 319 41.0421 315 4.2520 $N=10$ $N=20$ $N=30$ $N=40$ $k=30$ Iter. Time Iter. Time Iter. Time Iter. Time Proposed Alg. 3 1.3819 3 1.7834 3 1.6555 3 1.7517 Thong Alg. (1) 72 1.1548 142 2.6436 136 2.888 207 4.4416 Thong Alg. (2) 1771 30.2921 1325 28.4023 1132 28.6053 920 26.5714 Thong Alg. (3) 101 1.5058 90 1.4923 156 2.9515 162 3.6149 Gibali Alg. 203 17.1568 255 30.849 282 31.6244 303 35.2953
 $N=10$ $N=20$ $N=30$ $N=40$ $k=20$ Iter. Time Iter. Time Iter. Time Iter. Time Proposed Alg. 3 1.3843 3 1.7672 3 1.7564 4 2.2017 Thong Alg. (1) 76 1.2902 139 2.7111 111 2.1715 232 37.7743 Thong Alg. (2) 2136 36.6812 1561 30.7776 1370 31.8672 1160 4.0453 Thong Alg. (3) 86 1.1655 152 2.3615 148 2.4878 178 29.0789 Gibali Alg. 150 12.0085 235 20.3243 319 41.0421 315 4.2520 $N=10$ $N=20$ $N=30$ $N=40$ $k=30$ Iter. Time Iter. Time Iter. Time Iter. Time Proposed Alg. 3 1.3819 3 1.7834 3 1.6555 3 1.7517 Thong Alg. (1) 72 1.1548 142 2.6436 136 2.888 207 4.4416 Thong Alg. (2) 1771 30.2921 1325 28.4023 1132 28.6053 920 26.5714 Thong Alg. (3) 101 1.5058 90 1.4923 156 2.9515 162 3.6149 Gibali Alg. 203 17.1568 255 30.849 282 31.6244 303 35.2953
Example 1 Comparison: Proposed Alg. with different values $\gamma$
 $(N, k)$ $\gamma = 0.1$ $\gamma = 0.5$ $\gamma = 0.7$ $\gamma = 0.99$ $(20, 10)$ No. of Iterations 3 3 3 3 CPU (Time) 1.6337 1.4830 1.4773 1.3843 $(20, 20)$ No. of Iterations 3 3 4 3 CPU (Time) 1.4606 1.4876 2.3980 1.7672 $(20, 30)$ No. of Iterations 4 5 4 3 CPU (Time) 1.8664 0.85257 1.7597 1.7564 $(20, 40)$ No. of Iterations 4 3 4 4 CPU (Time) 1.6573 1.5935 1.8008 2.2017 $(30, 10)$ No. of Iterations 3 3 3 3 CPU (Time) 1.3060 1.3376 1.4359 1.3819 $(30, 20)$ No. of Iterations 3 4 3 3 CPU (Time) 1.4630 1.7306 1.5115 1.7834 $(30, 30)$ No. of Iterations 4 3 4 3 CPU (Time) 1.7102 1.6399 1.7931 1.6555 $(30, 40)$ No. of Iterations 5 3 4 3 CPU (Time) 2.7099 1.6589 2.3287 1.7517
 $(N, k)$ $\gamma = 0.1$ $\gamma = 0.5$ $\gamma = 0.7$ $\gamma = 0.99$ $(20, 10)$ No. of Iterations 3 3 3 3 CPU (Time) 1.6337 1.4830 1.4773 1.3843 $(20, 20)$ No. of Iterations 3 3 4 3 CPU (Time) 1.4606 1.4876 2.3980 1.7672 $(20, 30)$ No. of Iterations 4 5 4 3 CPU (Time) 1.8664 0.85257 1.7597 1.7564 $(20, 40)$ No. of Iterations 4 3 4 4 CPU (Time) 1.6573 1.5935 1.8008 2.2017 $(30, 10)$ No. of Iterations 3 3 3 3 CPU (Time) 1.3060 1.3376 1.4359 1.3819 $(30, 20)$ No. of Iterations 3 4 3 3 CPU (Time) 1.4630 1.7306 1.5115 1.7834 $(30, 30)$ No. of Iterations 4 3 4 3 CPU (Time) 1.7102 1.6399 1.7931 1.6555 $(30, 40)$ No. of Iterations 5 3 4 3 CPU (Time) 2.7099 1.6589 2.3287 1.7517
Example 1 Comparison: Proposed Alg. with different values $\mu$
 $(N, k)$ $\mu = 0.1$ $\mu = 0.5$ $\mu = 0.7$ $\mu = 0.99$ $(20, 10)$ No. of Iterations 3 3 4 3 CPU (Time) 1.4409 1.4632 2.6888 1.3843 $(20, 20)$ No. of Iterations 3 3 3 3 CPU (Time) 1.5248 1.4840 1.5217 1.7672 $(20, 30)$ No. of Iterations 4 3 5 3 CPU (Time) 1.9571 1.5852 1.9322 1.7564 $(20, 40)$ No. of Iterations 6 4 5 4 CPU (Time) 3.0365 1.8605 2.0718 2.2017 $(30, 10)$ No. of Iterations 3 3 3 3 CPU (Time) 1.3524 1.3416 1.3648 1.3819 $(30, 20)$ No. of Iterations 3 3 4 3 CPU (Time) 1.5265 1.5336 1.6929 1.7834 $(30, 30)$ No. of Iterations 5 5 3 3 CPU (Time) 2.9525 2.0816 1.6424 1.6555 $(30, 40)$ No. of Iterations 3 4 8 3 CPU (Time) 1.6958 2.0833 4.5199 1.7517
 $(N, k)$ $\mu = 0.1$ $\mu = 0.5$ $\mu = 0.7$ $\mu = 0.99$ $(20, 10)$ No. of Iterations 3 3 4 3 CPU (Time) 1.4409 1.4632 2.6888 1.3843 $(20, 20)$ No. of Iterations 3 3 3 3 CPU (Time) 1.5248 1.4840 1.5217 1.7672 $(20, 30)$ No. of Iterations 4 3 5 3 CPU (Time) 1.9571 1.5852 1.9322 1.7564 $(20, 40)$ No. of Iterations 6 4 5 4 CPU (Time) 3.0365 1.8605 2.0718 2.2017 $(30, 10)$ No. of Iterations 3 3 3 3 CPU (Time) 1.3524 1.3416 1.3648 1.3819 $(30, 20)$ No. of Iterations 3 3 4 3 CPU (Time) 1.5265 1.5336 1.6929 1.7834 $(30, 30)$ No. of Iterations 5 5 3 3 CPU (Time) 2.9525 2.0816 1.6424 1.6555 $(30, 40)$ No. of Iterations 3 4 8 3 CPU (Time) 1.6958 2.0833 4.5199 1.7517
Methods Parameters Choice for Comparison
 Proposed Alg. $\epsilon_n = \frac{1}{(n + 1)^2}$ $\theta = 0.5$ $l=0.01$ $\beta_n = \frac{1}{n + 1}$ $\gamma = 0.99$ $\mu = 0.99$ Gibali Alg. $\alpha_n = \frac{1}{1 + n}$ $l=0.01$ $\gamma = 0.99$ $\mu = 0.99$
 Proposed Alg. $\epsilon_n = \frac{1}{(n + 1)^2}$ $\theta = 0.5$ $l=0.01$ $\beta_n = \frac{1}{n + 1}$ $\gamma = 0.99$ $\mu = 0.99$ Gibali Alg. $\alpha_n = \frac{1}{1 + n}$ $l=0.01$ $\gamma = 0.99$ $\mu = 0.99$
Example 2: Prop. Alg. vs Gibali Alg. (Unaccel. Alg.)
 No. of Iterations CPU Time Prop. Alg. Gibali Alg. Prop. Alg. Gibali Alg. Case I 17 1712 0.001243 0.1244 Case II 17 1708 0.001518 0.1248 Case III 17 1713 0.001261 0.1276 Case IV 17 1729 0.001202 0.1297 Case V 17 1715 0.001272 0.1258 Case VI 18 1835 0.001339 0.1564
 No. of Iterations CPU Time Prop. Alg. Gibali Alg. Prop. Alg. Gibali Alg. Case I 17 1712 0.001243 0.1244 Case II 17 1708 0.001518 0.1248 Case III 17 1713 0.001261 0.1276 Case IV 17 1729 0.001202 0.1297 Case V 17 1715 0.001272 0.1258 Case VI 18 1835 0.001339 0.1564
Example 2 Comparison: Proposed Alg. with different values $\mu$
 $\mu = 0.1$ $\mu = 0.5$ $\mu = 0.7$ $\mu = 0.99$ Case I No. of Iterations 17 17 17 17 CPU (Time) 0.0011992 0.0012179 0.0013264 0.0012430 Case II No. of Iterations 17 17 17 17 CPU (Time) 0.0011457 0.0011586 0.0015604 0.0015181 Case III No. of Iterations 17 17 17 17 CPU (Time) 0.0011386 0.0014248 0.0012852 0.0012606 Case IV No. of Iterations 17 17 17 17 CPU (Time) 0.0010843 0.0010928 0.0011176 0.0012022 Case V No. of Iterations 17 17 17 17 CPU (Time) 0.0012491 0.0011169 0.0012293 0.0012719 Case VI No. of Iterations 18 18 18 18 CPU (Time) 0.0012431 0.0013496 0.0011613 0.0013392
 $\mu = 0.1$ $\mu = 0.5$ $\mu = 0.7$ $\mu = 0.99$ Case I No. of Iterations 17 17 17 17 CPU (Time) 0.0011992 0.0012179 0.0013264 0.0012430 Case II No. of Iterations 17 17 17 17 CPU (Time) 0.0011457 0.0011586 0.0015604 0.0015181 Case III No. of Iterations 17 17 17 17 CPU (Time) 0.0011386 0.0014248 0.0012852 0.0012606 Case IV No. of Iterations 17 17 17 17 CPU (Time) 0.0010843 0.0010928 0.0011176 0.0012022 Case V No. of Iterations 17 17 17 17 CPU (Time) 0.0012491 0.0011169 0.0012293 0.0012719 Case VI No. of Iterations 18 18 18 18 CPU (Time) 0.0012431 0.0013496 0.0011613 0.0013392
Example 2 Comparison: Proposed Alg. with different values $\gamma$
 $\gamma = 0.1$ $\gamma = 0.5$ $\gamma = 0.7$ $\gamma = 0.99$ Case I No. of Iterations 17 17 17 17 CPU (Time) 0.0013518 0.0012097 0.0011754 0.0012430 Case II No. of Iterations 17 17 17 17 CPU (Time) 0.0012701 0.0011233 0.0012382 0.0015181 Case III No. of Iterations 17 17 17 17 CPU (Time) 0.0011386 0.0014248 0.0012852 0.0012606 Case IV No. of Iterations 17 17 17 17 CPU (Time) 0.0011530 0.0013917 0.0015395 0.0012022 Case V No. of Iterations 17 17 17 17 CPU (Time) 0.0011413 0.0011319 0.0011286 0.0012719 Case VI No. of Iterations 17 17 18 18 CPU (Time) 0.0011094 0.0011839 0.0013550 0.0013392
 $\gamma = 0.1$ $\gamma = 0.5$ $\gamma = 0.7$ $\gamma = 0.99$ Case I No. of Iterations 17 17 17 17 CPU (Time) 0.0013518 0.0012097 0.0011754 0.0012430 Case II No. of Iterations 17 17 17 17 CPU (Time) 0.0012701 0.0011233 0.0012382 0.0015181 Case III No. of Iterations 17 17 17 17 CPU (Time) 0.0011386 0.0014248 0.0012852 0.0012606 Case IV No. of Iterations 17 17 17 17 CPU (Time) 0.0011530 0.0013917 0.0015395 0.0012022 Case V No. of Iterations 17 17 17 17 CPU (Time) 0.0011413 0.0011319 0.0011286 0.0012719 Case VI No. of Iterations 17 17 18 18 CPU (Time) 0.0011094 0.0011839 0.0013550 0.0013392
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