Self adaptive viscosity-type inertial extragradient algorithms for solving variational inequalities with applications
Abstract
In this paper, we introduce two new inertial extragradient algorithms with non-monotonic stepsizes for solving monotone and Lipschitz continuous variational inequality problems in real Hilbert spaces. Strong convergence theorems of the suggested iterative schemes are established without the prior knowledge of the Lipschitz constant of the mapping. Finally, some numerical examples are provided to illustrate the efficiency and advantages of the proposed algorithms and compare them with some related ones.
Keyword : variational inequality problem, optimal control problem, inertial subgradient extragradient method, inertial Tseng extragradient method, viscosity method
How to Cite
Tan, B., & Qin, X. (2022). Self adaptive viscosity-type inertial extragradient algorithms for solving variational inequalities with applications. Mathematical Modelling and Analysis, 27(1), 41–58. https://doi.org/10.3846/mma.2022.13846
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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L.O. Jolaoso, T.O. Alakoya, A. Taiwo and O.T. Mewomo. Inertial extragradient method via viscosity approximation approach for solving equilibrium problem in Hilbert space. Optimization, 70(2):387–412, 2021. https://doi.org/10.1080/02331934.2020.1716752
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R. Kraikaew and S. Saejung. Strong convergence of the halpern subgradient extragradient method for solving variational inequalities in Hilbert spaces. Journal of Optimization Theory and Applications, 163:399–412, 2014. https://doi.org/10.1007/s10957-013-0494-2
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J. Preininger and P.T. Vuong. On the convergence of the gradient projection method for convex optimal control problems with bang-bang solutions. Computational Optimization and Applications, 70:221–238, 2018. https://doi.org/10.1007/s10589-018-9981-6
S. Saejung and P. Yotkaew. Approximation of zeros of inverse strongly monotone mappings in Banach spaces. Nonlinear Analysis: Theory, Methods & Applications, 75(2):742–750, 2012. https://doi.org/10.1016/j.na.2011.09.005
D.R. Sahu, J.C. Yao, M. Verma and K.K. Shukla. Convergence rate analysis of proximal gradient methods with applications to composite minimization problems. Optimization, 70(1):75–100, 2021. https://doi.org/10.1080/02331934.2019.1702040
Y. Shehu and A. Gibali. New inertial relaxed method for solving split feasibilities. Optimization Letters, 15:2109–2126, 2021. https://doi.org/10.1007/s11590-020-01603-1
Y. Shehu and O. Iyiola. Strong convergence result for monotone variational inequalities. Numerical Algorithms, 76:259–282, 2017. https://doi.org/10.1007/s11075-016-0253-1
A. Taiwo, T.O. Alakoya and O.T. Mewomo. Halpern-type iterative process for solving split common fixed point and monotone variational inclusion problem between Banach spaces. Numerical Algorithms, 86:1359–1389, 2021. https://doi.org/10.1007/s11075-020-00937-2
A. Taiwo, L.O. Jolaoso and O.T. Mewomo. Inertial-type algorithm for solving split common fixed-point problem in Banach spaces. Journal of Scientific Computing, 86, 2021. https://doi.org/10.1007/s10915-020-01385-9
W. Takahashi. Nonlinear Functional Analysis. Yokohama Publishers, Yokohama, 2000.
W. Takahashi, C.F. Wen and J.-Ch. Yao. The shrinking projection method for a finite family of demimetric mappings with variational inequality problems in a Hilbert space. Fixed Point Theory, 19(1):407–420, 2018. https://doi.org/10.24193/fpt-ro.2018.1.32
B. Tan, J. Fan and S. Li. Self-adaptive inertial extragradient algorithms for solving variational inequality problems. Computational and Applied Mathematics, 40(19), 2021. https://doi.org/10.1007/s40314-020-01393-3
B. Tan, S. Xu and S. Li. Inertial shrinking projection algorithms for solving hierarchical variational inequality problems. Journal of Nonlinear and Convex Analysis, 21(4):871–884, 2020.
D.V. Thong and D.V. Hieu. Weak and strong convergence theorems for variational inequality problems. Numerical Algorithms, 78:1045–1060, 2018. https://doi.org/10.1007/s11075-017-0412-z
P. Tseng. A modified forward-backward splitting method for maximal monotone mappings. SIAM Journal on Control and Optimization, 38(2):431–446, 2000. https://doi.org/10.1137/S0363012998338806
P.T. Vuong and Y. Shehu. Convergence of an extragradient-type method for variational inequality with applications to optimal control problems. Numerical Algorithms, 81:269–291, 2019. https://doi.org/10.1007/s11075-018-0547-6
J. Yang and H. Liu. Strong convergence result for solving monotone variational inequalities in Hilbert space. Numerical Algorithms, 80:741–752, 2019. https://doi.org/10.1007/s11075-018-0504-4
J. Yang, H. Liu and Z. Liu. Modified subgradient extragradient algorithms for solving monotone variational inequalities. Optimization, 67(12):2247–2258, 2018. https://doi.org/10.1080/02331934.2018.1523404
T.O. Alakoya, L.O. Jolaoso and O.T. Mewomo. Modified inertial subgradient extragradient method with self adaptive stepsize for solving monotone variational inequality and fixed point problems. Optimization, 70(3):545–574, 2021. https://doi.org/10.1080/02331934.2020.1723586
T.O. Alakoya, A. Taiwo, O.T. Mewomo and Y.J. Cho. An iterative algorithm for solving variational inequality, generalized mixed equilibrium, convex minimization and zeros problems for a class of nonexpansive-type mappings. Ann Univ Ferrara, 67:1–31, 2021. https://doi.org/10.1007/s11565-020-00354-2
Q.H. Ansari, M. Islam and J.-Ch. Yao. Nonsmooth variational inequalities on Hadamard manifolds. Applicable Analysis, 99(2):340–358, 2020. https://doi.org/10.1080/00036811.2018.1495329
Y. Censor, A. Gibali and S. Reich. Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space. Optimization Methods and Software, 26(4–5):827–845, 2011. https://doi.org/10.1080/10556788.2010.551536
T.H. Cuong, J.-Ch. Yao and N.D. Yen. Qualitative properties of the minimum sum-of-squares clustering problem. Optimization, 69(9):2131–2154, 2020. https://doi.org/10.1080/02331934.2020.1778685
A. Gibali and D.V. Hieu. A new inertial double-projection method for solving variational inequalities. Journal of Fixed Point Theory and Applications, 21(4), 2019. https://doi.org/10.1007/s11784-019-0726-7
D.V. Hieu, J.J. Strodiot and L.D. Muu. Strongly convergent algorithms by using new adaptive regularization parameter for equilibrium problems. Journal of Computational and Applied Mathematics, 376, 2020. https://doi.org/10.1016/j.cam.2020.112844
C. Izuchukwu, A.A. Mebawondu and O.T. Mewomo. A new method for solving split variational inequality problems without co-coerciveness. Journal of Fixed Point Theory and Applications, 22(98), 2020. https://doi.org/10.1007/s11784-020-00834-0
C. Izuchukwu, G.N. Ogwo and O.T. Mewomo. An inertial method for solving generalized split feasibility problems over the solution set of monotone variational inclusions. Optimization, pp. 1–29, 2020. https://doi.org/10.1080/02331934.2020.1808648
L.O. Jolaoso, T.O. Alakoya, A. Taiwo and O.T. Mewomo. Inertial extragradient method via viscosity approximation approach for solving equilibrium problem in Hilbert space. Optimization, 70(2):387–412, 2021. https://doi.org/10.1080/02331934.2020.1716752
L.O. Jolaoso, A. Taiwo, T.O. Alakoya and O.T. Mewomo. A strong convergence theorem for solving pseudo-monotone variational inequalities using projection methods. Journal of Optimization Theory and Applications, 185:744–766, 2020. https://doi.org/10.1007/s10957-020-01672-3
A.A. Khan, S. Migorski and M. Sama. Inverse problems for multivalued quasi variational inequalities and noncoercive variational inequalities with noisy data. Optimization, 68(10):1897–1931, 2019. https://doi.org/10.1080/02331934.2019.1604706
G.M. Korpelevich. The extragradient method for finding saddle points and other problems. Ekonomika i Matematicheskie Metody, 12(4):747–756, 1976.
R. Kraikaew and S. Saejung. Strong convergence of the halpern subgradient extragradient method for solving variational inequalities in Hilbert spaces. Journal of Optimization Theory and Applications, 163:399–412, 2014. https://doi.org/10.1007/s10957-013-0494-2
H. Liu and J. Yang. Weak convergence of iterative methods for solving quasimonotone variational inequalities. Computational Optimization and Applications, 77:491–508, 2020. https://doi.org/10.1007/s10589-020-00217-8
L. Liu. A hybrid steepest descent method for solving split feasibility problems involving nonexpansive mappings. Journal of Nonlinear and Convex Analysis, 20(3):471–488, 2019.
A. Pietrus, T. Scarinci and V.M. Veliov. High order discrete approximations to Mayer’s problems for linear systems. SIAM Journal on Control and Optimization, 56(1):102–119, 2018. https://doi.org/10.1137/16M1079142
J. Preininger and P.T. Vuong. On the convergence of the gradient projection method for convex optimal control problems with bang-bang solutions. Computational Optimization and Applications, 70:221–238, 2018. https://doi.org/10.1007/s10589-018-9981-6
S. Saejung and P. Yotkaew. Approximation of zeros of inverse strongly monotone mappings in Banach spaces. Nonlinear Analysis: Theory, Methods & Applications, 75(2):742–750, 2012. https://doi.org/10.1016/j.na.2011.09.005
D.R. Sahu, J.C. Yao, M. Verma and K.K. Shukla. Convergence rate analysis of proximal gradient methods with applications to composite minimization problems. Optimization, 70(1):75–100, 2021. https://doi.org/10.1080/02331934.2019.1702040
Y. Shehu and A. Gibali. New inertial relaxed method for solving split feasibilities. Optimization Letters, 15:2109–2126, 2021. https://doi.org/10.1007/s11590-020-01603-1
Y. Shehu and O. Iyiola. Strong convergence result for monotone variational inequalities. Numerical Algorithms, 76:259–282, 2017. https://doi.org/10.1007/s11075-016-0253-1
A. Taiwo, T.O. Alakoya and O.T. Mewomo. Halpern-type iterative process for solving split common fixed point and monotone variational inclusion problem between Banach spaces. Numerical Algorithms, 86:1359–1389, 2021. https://doi.org/10.1007/s11075-020-00937-2
A. Taiwo, L.O. Jolaoso and O.T. Mewomo. Inertial-type algorithm for solving split common fixed-point problem in Banach spaces. Journal of Scientific Computing, 86, 2021. https://doi.org/10.1007/s10915-020-01385-9
W. Takahashi. Nonlinear Functional Analysis. Yokohama Publishers, Yokohama, 2000.
W. Takahashi, C.F. Wen and J.-Ch. Yao. The shrinking projection method for a finite family of demimetric mappings with variational inequality problems in a Hilbert space. Fixed Point Theory, 19(1):407–420, 2018. https://doi.org/10.24193/fpt-ro.2018.1.32
B. Tan, J. Fan and S. Li. Self-adaptive inertial extragradient algorithms for solving variational inequality problems. Computational and Applied Mathematics, 40(19), 2021. https://doi.org/10.1007/s40314-020-01393-3
B. Tan, S. Xu and S. Li. Inertial shrinking projection algorithms for solving hierarchical variational inequality problems. Journal of Nonlinear and Convex Analysis, 21(4):871–884, 2020.
D.V. Thong and D.V. Hieu. Weak and strong convergence theorems for variational inequality problems. Numerical Algorithms, 78:1045–1060, 2018. https://doi.org/10.1007/s11075-017-0412-z
P. Tseng. A modified forward-backward splitting method for maximal monotone mappings. SIAM Journal on Control and Optimization, 38(2):431–446, 2000. https://doi.org/10.1137/S0363012998338806
P.T. Vuong and Y. Shehu. Convergence of an extragradient-type method for variational inequality with applications to optimal control problems. Numerical Algorithms, 81:269–291, 2019. https://doi.org/10.1007/s11075-018-0547-6
J. Yang and H. Liu. Strong convergence result for solving monotone variational inequalities in Hilbert space. Numerical Algorithms, 80:741–752, 2019. https://doi.org/10.1007/s11075-018-0504-4
J. Yang, H. Liu and Z. Liu. Modified subgradient extragradient algorithms for solving monotone variational inequalities. Optimization, 67(12):2247–2258, 2018. https://doi.org/10.1080/02331934.2018.1523404