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A note on the paper "Sufficient optimality conditions using convexifactors for optimistic bilevel programming problem"
doi: 10.3934/jimo.2022060
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Inertial iterative method for solving variational inequality problems of pseudo-monotone operators and fixed point problems of nonexpansive mappings in Hilbert spaces

 1 Department of Mathematics, Zhejiang Normal University, Jinhua 321004, Zhejiang, China 2 Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China 3 Department of Mathematics, Luoyang Normal University, Luoyang, 471022, China

*Corresponding author: Yuanheng Wang

Received  July 2021 Revised  March 2022 Early access April 2022

Fund Project: This work was supported by the NSF of China (12171435, 11971216)

In this paper, we propose a new inertial viscosity iterative algorithm for solving the variational inequality problem with a pseudo-monotone operator and the fixed point problem involving a nonexpansive mapping in real Hilbert spaces. The advantage of the proposed algorithm is that it can work without the prior knowledge of the Lipschitz constant of the mapping. The strong convergence of the sequence generated by the proposed algorithm is proved under some suitable assumptions imposed on the parameters. Some numerical experiments are given to support our main results.

Citation: 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, doi: 10.3934/jimo.2022060
References:

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References:
The behavior of our Algorithm 3.1 in Example 4.1 ($m = 200$)
The behavior of our Algorithm 3.1 in Example 4.2 ($x_{0} = x_{1} = 10\exp(t)$)
The behavior of our Algorithm 3.1 in Example 4.3 ($m = 500000$)
Numerical results of all algorithms with different dimensions in Example 4.1
 Algorithms $m=20$ $m=50$ $m=100$ $m=200$ $D_{n}$ CPU $D_{n}$ CPU $D_{n}$ CPU $D_{n}$ CPU Our Alg. 3.1 4.47E-05 0.0303 1.70E-04 0.0316 3.17E-04 0.0707 5.00E-04 0.1119 CTC Alg. 3.1 3.36E-04 0.0227 9.95E-04 0.0227 1.69E-03 0.0250 2.64E-03 0.0273 GTT Alg. 3.1 2.80E-03 0.0310 6.86E-03 0.0361 1.08E-02 0.1044 1.63E-02 0.1393 GTT Alg. 3.2 3.38E-03 0.0483 9.78E-03 0.0379 1.64E-02 0.0704 2.56E-02 0.1027
 Algorithms $m=20$ $m=50$ $m=100$ $m=200$ $D_{n}$ CPU $D_{n}$ CPU $D_{n}$ CPU $D_{n}$ CPU Our Alg. 3.1 4.47E-05 0.0303 1.70E-04 0.0316 3.17E-04 0.0707 5.00E-04 0.1119 CTC Alg. 3.1 3.36E-04 0.0227 9.95E-04 0.0227 1.69E-03 0.0250 2.64E-03 0.0273 GTT Alg. 3.1 2.80E-03 0.0310 6.86E-03 0.0361 1.08E-02 0.1044 1.63E-02 0.1393 GTT Alg. 3.2 3.38E-03 0.0483 9.78E-03 0.0379 1.64E-02 0.0704 2.56E-02 0.1027
Numerical results of all algorithms with different initial values in Example 4.2
 Algorithms $x_{1}=10t^{3}$ $x_{1}=10\sin(2t)$ $x_{1}=10\log(t)$ $x_{1}=10\exp(t)$ $D_{n}$ CPU $D_{n}$ CPU $D_{n}$ CPU $D_{n}$ CPU Our Alg. 3.1 1.47E-15 37.7392 4.04E-15 38.5818 7.26E-15 39.9932 3.16E-15 44.6711 CTC Alg. 3.1 3.16E-13 22.8652 2.55E-13 24.3823 3.36E-12 25.0381 4.39E-12 30.0801 GTT Alg. 3.1 9.04E-06 33.7345 1.54E-05 34.5784 2.31E-05 36.9545 2.01E-05 45.1488 GTT Alg. 3.2 6.25E-11 33.5898 7.40E-10 36.0191 1.04E-09 37.5059 1.06E-09 44.7099
 Algorithms $x_{1}=10t^{3}$ $x_{1}=10\sin(2t)$ $x_{1}=10\log(t)$ $x_{1}=10\exp(t)$ $D_{n}$ CPU $D_{n}$ CPU $D_{n}$ CPU $D_{n}$ CPU Our Alg. 3.1 1.47E-15 37.7392 4.04E-15 38.5818 7.26E-15 39.9932 3.16E-15 44.6711 CTC Alg. 3.1 3.16E-13 22.8652 2.55E-13 24.3823 3.36E-12 25.0381 4.39E-12 30.0801 GTT Alg. 3.1 9.04E-06 33.7345 1.54E-05 34.5784 2.31E-05 36.9545 2.01E-05 45.1488 GTT Alg. 3.2 6.25E-11 33.5898 7.40E-10 36.0191 1.04E-09 37.5059 1.06E-09 44.7099
Numerical results of all algorithms with different dimensions in Example 4.3
 Algorithms $m=500$ $m=5000$ $m=50000$ $m=500000$ $D_{n}$ CPU $D_{n}$ CPU $D_{n}$ CPU $D_{n}$ CPU Our Alg. 3.1 7.13E-57 0.0249 8.76E-57 0.1079 3.77E-57 0.4058 8.89E-57 13.8430 CDP Alg. 3.1 3.97E-27 0.0406 7.89E-27 0.1274 7.25E-27 0.5290 4.78E-26 13.9558 TSI Alg. 3 8.38E-13 0.0318 7.96E-13 0.1270 8.17E-13 0.4180 6.62E-13 15.0426 RTDLD Alg. 4 4.72E-10 0.0312 3.07E-07 0.1132 1.64E-03 0.4540 2.59E-02 19.2562
 Algorithms $m=500$ $m=5000$ $m=50000$ $m=500000$ $D_{n}$ CPU $D_{n}$ CPU $D_{n}$ CPU $D_{n}$ CPU Our Alg. 3.1 7.13E-57 0.0249 8.76E-57 0.1079 3.77E-57 0.4058 8.89E-57 13.8430 CDP Alg. 3.1 3.97E-27 0.0406 7.89E-27 0.1274 7.25E-27 0.5290 4.78E-26 13.9558 TSI Alg. 3 8.38E-13 0.0318 7.96E-13 0.1270 8.17E-13 0.4180 6.62E-13 15.0426 RTDLD Alg. 4 4.72E-10 0.0312 3.07E-07 0.1132 1.64E-03 0.4540 2.59E-02 19.2562

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