100 101 102 103 104 105
|| e n || 2
10-5 100 105
Proposed Alg. 1 Bot Alg.
Malitsky Alg.
Shehu Alg.
Figure 9.Example2: µ=φ=γ=0.7 and λ0=1.
Number of iterations
100 101 102 103 104
|| e n || 2
10-5 100 105
Proposed Alg. 1 Bot Alg.
Malitsky Alg.
Shehu Alg.
Figure 10.Example2: µ=φ=γ=0.999 and λ0=1.
Number of iterations
100 101 102
|| e n || 2
10-5 10-4 10-3 10-2 10-1 100 101 102
Proposed Alg. 3: µ = 0.1 Proposed Alg. 3: µ = 0.3 Proposed Alg. 3: µ = 0.7 Proposed Alg. 3: µ = 0.999
Figure 11.Example2: λ=1.
Number of iterations
100 101 102
|| e n || 2
10-5 10-4 10-3 10-2 10-1 100 101 102
Proposed Alg. 3: λ0= 0.1 Proposed Alg. 3: λ0= 1 Proposed Alg. 3: λ0= 5 Proposed Alg. 3: λ0= 10
Figure 12.Example2: µ=0.999.
Table 5.Comparison of proposed Algorithm3and the subgradient-extragradient method (SEM) (2) for Example3.
m=10 m=20 m=30 m=40
k=30 Iter. CPU Time Iter. CPU Time Iter. CPU Time Iter. CPU Time
Proposed
Algorithm3 157 2.7327 162 3.9759 144 4.4950 128 4.8193
SEM (2) 3785 64.8752 13,980 243.9019 18,994 345.3686 30,777 567.8440
m=10 m=20 m=30 m=40
k=50 Iter. CPU Time Iter. CPU Time Iter. CPU Time Iter. CPU Time
Proposed
Algorithm3 185 4.0691 173 4.3798 128 4.8817 130 6.2658
SEM (2) 4176 77.6893 8645 150.4267 21,262 381.0991 30,956 561.4559
Number of iterations
100 101 102 103 104
|| e n || 2
10-3 10-2 10-1 100 101
Proposed Alg. 3 SEM
Figure 13.Example3: k=30 and m=10.
Number of iterations
100 101 102 103 104 105
|| e n || 2
10-3 10-2 10-1 100 101
Proposed Alg. 3 SEM
Figure 14.Example3: k=30 and m=20.
Number of iterations
100 101 102 103 104 105
|| e n || 2
10-3 10-2 10-1 100 101
Proposed Alg. 3 SEM
Figure 15.Example3: k=30 and m=30.
Number of iterations
100 101 102 103 104 105
|| e n || 2
10-3 10-2 10-1 100 101
Proposed Alg. 3 SEM
Figure 16.Example3: k=30 and m=40.
Number of iterations
100 101 102 103 104
|| e n || 2
10-3 10-2 10-1 100 101
Proposed Alg. 3 SEM
Figure 17.Example3: k=50 and m=10.
Number of iterations
100 101 102 103 104
|| e n || 2
10-3 10-2 10-1 100 101
Proposed Alg. 3 SEM
Figure 18.Example3: k=50 and m=20.
Number of iterations
100 101 102 103 104 105
|| e n || 2
10-3 10-2 10-1 100 101
Proposed Alg. 3 SEM
Figure 19.Example3: k=50 and m=30.
Number of iterations
100 101 102 103 104 105
|| e n || 2
10-3 10-2 10-1 100 101
Proposed Alg. 3 SEM
Figure 20.Example3: k=50 and m=40.
Number of iterations
100 101 102 103
|| e n || 2
10-3 10-2 10-1 100 101
Proposed Alg. 3: m = 10 Proposed Alg. 3: m = 20 Proposed Alg. 3: m = 30 Proposed Alg. 3: m = 40
Figure 21.Example3: k=30.
Number of iterations
100 101 102 103
|| e n || 2
10-3 10-2 10-1 100 101
Proposed Alg. 3: m = 10 Proposed Alg. 3: m = 20 Proposed Alg. 3: m = 30 Proposed Alg. 3: m = 40
Figure 22.Example3: k=50.
Table 6. Comparison of proposed Algorithm3 and the extragradient method (EGM) [16] for Example4with λ0=1 and µ=0.1.
Proposed Algorithm3 EGM [16]
x1 No. of Iter. CPU Time No. of Iter. CPU Time
(2, 1)T 13 5.9390×10−4 62 1.8851×10−3
(1, 2)T 33 5.2230×10−4 62 1.9666×10−3
(1.5, 1.5)T 12 4.7790×10−4 14 3.955×10−4
(1.25, 1.75)T 13 6.7980×10−4 58 1.9138×10−3
Table 7.Proposed Algorithm3for Example4with λ0=5 and µ=0.999.
Proposed Algorithm3
x1 No. of Iter. CPU Time
(2, 1)T 18 7.5980×10−4
(1, 2)T 15 6.0910×10−4
(1.5, 1.5)T 13 5.5840×10−4
(1.25, 1.75)T 16 6.9110×10−4
Table 8.Example4: proposed Algorithm3with x1= (2, 1)Tfor different µ and λ0values.
λ0=5
µ=0.1 µ=0.3 µ=0.7 µ=0.999
No. of Iter. 14 14 16 18
CPU Time 6.8600×10−4 7.1170×10−4 8.1080×10−4 8.6850×10−4 µ=0.999
λ0=0.1 λ0=1 λ0=5 λ0=10
No. of Iter. 12 18 18 18
CPU Time 5.6360×10−4 9.3920×10−4 9.0230×10−4 8.9310×10−4
Number of iterations
100 101 102
|| e n || 2
10-4 10-3 10-2 10-1 100 101
Proposed Alg. 1 EGM Alg.
Figure 23.Example4: k=50 and m=10.
Number of iterations
100 101 102
|| e n || 2
10-4 10-3 10-2 10-1 100 101
Proposed Alg. 1 EGM Alg.
Figure 24.Example4: k=50 and m=20.
Number of iterations
100 101 102
|| e n || 2
10-4 10-3 10-2 10-1 100 101
Proposed Alg. 1 EGM Alg.
Figure 25.Example4: k=50 and m=30.
Number of iterations
100 101 102
|| e n || 2
10-4 10-3 10-2 10-1 100 101
Proposed Alg. 1 EGM Alg.
Figure 26.Example4: k=50 and m=40.
Number of iterations
100 101 102
|| e n || 2
10-4 10-3 10-2 10-1 100 101
Proposed Alg. 3: x1= (2, 1)T Proposed Alg. 3: x1= (1, 2)T Proposed Alg. 3: x1= (1.5, 1.5)T Proposed Alg. 3: x1= (1.25, 1.75)T
Figure 27.Example4: λ0=1 and µ=0.1.
Number of iterations
100 101 102
|| e n || 2
10-5 10-4 10-3 10-2 10-1 100 101
Proposed Alg. 3: x1= (2, 1)T Proposed Alg. 3: x1= (1, 2)T Proposed Alg. 3: x1= (1.5, 1.5)T Proposed Alg. 3: x1= (1.25, 1.75)T
Figure 28.Example4: λ0=5 and µ=0.999.
Number of iterations
100 101 102
|| e n || 2
10-4 10-3 10-2 10-1 100 101
Proposed Alg. 3: µ = 0.1 Proposed Alg. 3: µ = 0.3 Proposed Alg. 3: µ = 0.7 Proposed Alg. 3: µ = 0.999
Figure 29.Example4: λ0=5 and x1= (2, 1)T.
Number of iterations
100 101 102
|| e n || 2
10-4 10-3 10-2 10-1 100 101
Proposed Alg. 3: λ0= 0.1 Proposed Alg. 3: λ0= 1 Proposed Alg. 3: λ0= 5 Proposed Alg. 3: λ0= 10
Figure 30.Example4: µ=0.999 and x1= (2, 1)T.
Table 9.Example5comparison: proposed Algorithm3, Bot Algorithm1, Malitsky Algorithm2, and Shehu Alg. [37] with λ0=1 and µ=0.9.
Proposed Algorithm3 Bot Algorithm1 Malitsky Algorithm2 Shehu Alg. [37]
x1 Iter. CPU Time Iter. CPU Time Iter. CPU Time Iter. CPU Time
1
12(t2−2t+1) 23 5.1433×10−3 2159 0.23341 371 3.7625×10−2 37,135 10.8377
1
9etsin(t) 18 3.2111×10−3 1681 0.18084 374 3.9159×10−2 70,741 32.5843
1
21t2cos(t) 14 2.4545×10−3 4344 0.48573 373 3.8413×10−2 17,741 5.6272
1
7(3t−2)et 43 8.7538×10−3 2774 0.29515 351 3.7544×10−2 28,758 7.3424
Table 10.Example5: proposed Algorithm3with x1=t2−2t+112 for different µ and λ0values.
µ=0.9
λ0=0.1 λ0=1 λ0=2 λ0=3
No. of Iter. 167 23 11 8
CPU Time 3.4828×10−2 4.2089×10−3 2.2288×10−3 1.3899×10−3 λ0=2
µ=0.1 µ=0.3 µ=0.7 µ=0.999
No. of Iter. 11 11 11 11
CPU Time 2.0615×10−3 2.1237×10−3 1.9886×10−3 2.0384×10−3
Number of iterations
100 101 102 103 104 105
|| e n || 2
10-5 10-4 10-3 10-2 10-1 100 101
Proposed Alg. 1 Bot Alg.
Malitsky Alg.
Shehu Alg.
Figure 31.Example5: λ0=1, µ=0.9 and x1=121(t2−2t+1).
Number of iterations
100 101 102 103 104 105
|| e n || 2
10-5 10-4 10-3 10-2 10-1 100 101
Proposed Alg. 1 Bot Alg.
Malitsky Alg.
Shehu Alg.
Figure 32.Example5: λ0=1, µ=0.9 and x1=19etsin(t).
Number of iterations
100 101 102 103 104 105
|| e n || 2
10-5 10-4 10-3 10-2 10-1 100
Proposed Alg. 1 Bot Alg.
Malitsky Alg.
Shehu Alg.
Figure 33.Example5: λ0=1, µ=0.9 and x1=211t2cos(t).
Number of iterations
100 101 102 103 104 105
|| e n || 2
10-5 10-4 10-3 10-2 10-1 100 101
Proposed Alg. 1 Bot Alg.
Malitsky Alg.
Shehu Alg.
Figure 34.Example5: λ0=1, µ=0.9 and x1=17(3t−2)et.
Number of iterations
100 101 102 103
|| e n || 2
10-5 10-4 10-3 10-2 10-1 100
Proposed Alg. 3: λ0= 0.1 Proposed Alg. 3: λ0= 1 Proposed Alg. 3: λ0= 2 Proposed Alg. 3: λ0= 3
Figure 35.Example5: µ=0.9.
Number of iterations
100 101 102
|| e n || 2
10-5 10-4 10-3 10-2 10-1 100
Proposed Alg. 3: µ = 0.1 Proposed Alg. 3: µ = 0.3 Proposed Alg. 3: µ = 0.7 Proposed Alg. 3: µ = 0.999
Figure 36.Example5: λ0=2.
5. Discussion
The weak convergence analysis of the reflected subgradient-extragradient method for variational inequalities in real Hilbert spaces is obtained in this paper. We provide and intensive numerical illustration and comparison with related works for several applications such as tomography reconstruction and Nash–Cournot oligopolistic equilibrium models.
Our result is one of the few results on the subgradient-extragradient method with the reflected step in the literature. Our next project is to modify our results to bilevel variational inequalities.
Author Contributions: Writing—review and editing, A.G., O.S.I., L.A. and Y.S. All authors con-tributed equally to this work which included mathematical theory and analysis and code implemen-tation. All authors have read and agreed to the published version of the manuscript.
Funding:This research received no external funding.
Institutional Review Board Statement:Not applicable.
Informed Consent Statement:Not applicable.
Data Availability Statement:The study does not report any data.
Conflicts of Interest:The authors declare no conflict of interest.
Abbreviations
The following abbreviations are used in this manuscript:
VI variational inequality problem EGM extragradient method
SEGM subgradient-extragradient method References
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