Acknowledgement
This work was supported by the Basic Science Research Program through the National Research Foundation(NRF) Grant funded by Ministry of Education of the republic of Korea (2018R1D1A1B07045427).
References
- F. Alvarez, Weak convergence of a relaxed and inertial hybrid projection-proximal point algorithm for maximal monotone operators in Hilbert space, SIAM J. Optim., 14 (2004), 773-782. https://doi.org/10.1137/S1052623403427859
- P.N. Anh and Q.H. Ansari, Auxiliary problem technique for Hierarchical equilibrium problems, J. Optim. Theory Appl., 188(3) (2021), 882-912. https://doi.org/10.1007/s10957-021-01814-1
- P.N. Anh, N.D. Hien, N.X. Phuong and V.T. Ngoc, Parallel subgradient methods for variational inequalities involving nonexpansive mappings, Appl. Anal., (2019). Doi: 10.1080/00036811.2019.1584288.
- P.N. Anh and N.V. Hong, New projection methods for solving equilibrium problems over the fixed point sets, Optim. Lett., (2020). Doi: 10.1007/s11590-020-01625-9.
- P.N. Anh, J.K. Kim and L.D. Muu, An extragradient method for solving bilevel variational inequalities, J. Glob. Optim., 52 (2012), 627-639. https://doi.org/10.1007/s10898-012-9870-y
- P.N. Anh, H.T.C. Thach and J.K. Kim, Proximal-like subgradient methods for solving multi-valued variational inequalities, Nonlinear Funct. Anal. Appl., 25(3) (2020), 437-451. doi.org/10.22771/nfaa.2020.25.03.03.
- P.N. Anh, T.V. Thang and H.T.C. Thach, Halpern projection methods for solving pseudomonotone multivalued variational inequalities in Hilbert spaces, Numer. Algor., (2020). Doi: 10.1007/s11075-020-00968-9.
- P.N. Anh and H.A. Le Thi, New subgradient extragradient methods for solving monotone bilevel equilibrium problems, Optim., 68(1) (2019), 2097-2122.
- K. Aoyama and Y. Kimura, Strong convergence theorems for strongly nonexpansive sequences. Appl. Math. Comput., 217 (2011), 7537-7545. https://doi.org/10.1016/j.amc.2011.01.092
- K. Aoyama, F. Kohsaka and W. Takahashi, Strong convergence theorems by shrinking and hybrid projection methods for relatively nonexpansive mappings in Banach spaces, Proc. the 5th Int. Conference on Nonlinear Anal. Convex Anal., J. Nonl. Convex Anal., (2009) 7-26.
- A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imag. Sc., 2 (2009), 183-202. https://doi.org/10.1137/080716542
- R.I. Bot, E.R. Csetnek and S.C. Laszlo, An inertial forward-backward algorithm for the minimization of the sum of two nonconvex functions, EURO J. Comput. Optim., 4 (2015), 3-25. https://doi.org/10.1007/s13675-015-0045-8
- L. Bussaban, S. Suantai and A. Kaewkhao, A parallel inertial S-iteration forward-backward algorithm for regression and classification problems, Carpathian J. Math., 36 (2020), 35-44. https://doi.org/10.37193/CJM.2020.01.04
- L-C. Ceng, Q.H. Ansari and J-C. Yao, Mann-type steepest-descent and modified hybrid steepest-descent methods for variational inequalities in Banach spaces, Num. Funct. Anal. Optim., 29(9-10) (2008), 987-1033. https://doi.org/10.1080/01630560802418391
- L-C. Ceng, C. Lee and J-C. Yao, Strong weak convergence theorems of implicit hybrid steepest-descent methods for variational inequalities, Taiwanese J. Math., 12(1) (2008), 227-244. https://doi.org/10.11650/twjm/1500602499
- L-C. Ceng and M. Shang, Hybrid inertial subgradient extragradient methods for variational inequalities and fixed point problems involving asymptotically nonexpansive mappings, Optim., (2019). Doi: 10.1080/02331934.2019.1647203.
- Z. Chbani and H. Riahi, Weak and strong convergence of an inertial proximal method for solving Ky Fan minimax inequalities, Optim. Lett., 7 (2013), 185-206. https://doi.org/10.1007/s11590-011-0407-y
- X.P. Ding, Y.C. Lin and J-C. Yao, Three-step relaxed hybrid steepest-descent methods for variational inequalities, Appl. Math. Mech., 28 (2007), 1029-1036. https://doi.org/10.1007/s10483-007-0805-x
- F. Facchinei and J.S. Pang, Finite-dimensional variational inequalities and complementary problems, Springer, NewYork, 2003.
- J.K. Kim, A.H. Dar and Salahuddin, Existence theorems for the generalized relaxed pseudomonotone variational inequalities, Nonlinear Funct. Anal. Appl., 25(1) (2020), 25-34. doi.org/10.22771/nfaa.2020.25.01.03.
- D. Kinderlehrer and G. Stampacchia, An introduction to variational inequalities and their applications, Academic Press, 1980.
- I.V. Konnov, Combined relaxation methods for variational inequalities, Springer-Verlag, Berlin, 2000.
- G.M. Korpelevich, Extragradientmethod for finding saddle points and other problems, Matecon 12 (1976), 747-756.
- F. Liu and M.Z. Nashed, Regularization of nonlinear ill-posed variational inequalities and convergence rates, Set-Valued Anal., 6 (1998), 313-344. https://doi.org/10.1023/A:1008643727926
- D.A. Lorenz and T. Pock, An inertial forward-backward algorithm for monotone inclusions, J. Math. Imaging Vis., 51 (2015), 311-325. https://doi.org/10.1007/s10851-014-0523-2
- P.E. Mainge, A hybrid extragradient-viscosity method for monotone operators and fixed point problems, SIAM J. Control Optim., 47 (2008), 1499-1515. https://doi.org/10.1137/060675319
- P. Marcotte, Network design problem with congestion effects: A case of bilevel programming, Math. Progr., 34(2) (1986), 142-162. https://doi.org/10.1007/BF01580580
- K. Muangchoo, A viscosity type projection method for solving pseudomonotone variational inequalities, Nonlinear Funct. Anal. Appl., 26(2) (2021), 347-371. doi.org/10.22771/nfaa.2021.26.02.08.
- B.T. Polyak, Some methods of speeding up the convergence of iteration methods. USSR Comput. Math. Math. Phys., 4(5) (1964), 1-17. https://doi.org/10.1016/0041-5553(64)90137-5
- T. Ram, J.K. Kim and R. Kour, On Optimal Solutions of Well-posed Problems and Variational Inequalities, Nonlinear Funct. Anal. Appl., 24(4) (2021), 25-34. doi.org/10.22771/nfaa.2021.26.04.08.
- S. Saejung and P. Yotkaew, Approximation of zeros of inverse strongly monotone operators in Banach spaces, Nonlinear Anal., 75 (2012), 724-750.
- M.V. Solodov and P. Tseng, Modified projection-type methods for monotone variational inequalities, SIAM J. Control Optim., 34 (1996), 1814-1830. https://doi.org/10.1137/S0363012994268655
- J. Tang, J. Zhu, S.S. Chang, M. Liu and X. Li, A new modified proximal point algorithm for a finite family of minimization problem and fixed point for a finite family of demicontractive mappings in Hadamard spaces, Nonlinear Funct. Anal. Appl., 25(3) (2020), 563-577. doi.org/10.22771/nfaa.2020.25.03.11.
- M.H. Xu T.H. Kim, Convergence of hybrid steepest-descent methods for variational inequalities, J. Optim. Theory Appl., 119 (2003), 185-201. https://doi.org/10.1023/B:JOTA.0000005048.79379.b6
- M.H. Xu, M. Li and C.C.M. Yang, Neural networks for a class of bilevel variational inequalities, J. Glob. Optim., 44 (2009), 535-552. https://doi.org/10.1007/s10898-008-9355-1
- I. Yamada, The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings, Stud. Comput. Math., 8 (2001), 473-504. https://doi.org/10.1016/S1570-579X(01)80028-8
- I. Yamada and N. Ogura, Hybrid steepest descent method for the variational inequality problem over the the fixed point set of certain quasi-nonexpansive mappings, Numer. Funct. Anal. Optim., 25 (2004), 619-655. https://doi.org/10.1081/NFA-200045815
- Y. Yao and M.A. Noor, Strong convergence of the modified hybrid steepest-descent methods for general variational inequalities, J. Appl. Math. comput., 24(1-2) (2007), 179-190. https://doi.org/10.1007/BF02832309
- Y. Yao, M.A. Noor, R. Chen and Y.C. Liou, Strong convergence of three-step relaxed hybrid steepest-descent methods for variational inequalities, Appl. Math. Comput., 201 (2008), 175-183. https://doi.org/10.1016/j.amc.2007.12.011
- L.C. Zeng Q.H. Ansari and S.Y. Wu, Strong convergence theorems of relaxed hybrid steepest-descent methods for variational inequalities, Taiwanese J. Math., 10(1) (2006), 13-29. https://doi.org/10.11650/twjm/1500403796
- L.C. Zeng, N.C. Wong and J-C. Yao, Convergence of hybrid steepest-descent methods for generalized variational inequalities, Acta Mathematica Sinica, 22(1) (2006), 1-12.
- L.C. Zeng, N.C. Wong and J-C. Yao, Convergence analysis of modified hybrid steepest-descent methods with variable parameters for variational inequalities, J. Optim. Theory Appl., 132(1) (2007), 51-69. https://doi.org/10.1007/s10957-006-9068-x
- L.C. Zeng and J-C. Yao, Two step relaxed hybrid steepest-descent methods for variational inequalities, J. Inequal. Appl., 2008 (2008), 598-632.