An inertial Tseng algorithm for solving quasimonotone variational inequality and fixed point problem in Hilbert spaces. (English) Zbl 07924066
Summary: In this paper, we propose an inertial method for solving a common solution to fixed point and Variational Inequality Problem in Hilbert spaces. Under some standard and suitable assumptions on the control parameters, we prove that the sequence generated by the proposed algorithm converges strongly to an element in the solution set of Variational Inequality Problem associated with a quasimonotone operator which is also solution to a fixed point problem for a demimetric mapping. Finally, we give some numerical experiments for supporting our main results and also compare with some earlier announced methods in the literature.
MSC:
| 47H06 | Nonlinear accretive operators, dissipative operators, etc. |
| 47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |
| 47J05 | Equations involving nonlinear operators (general) |
| 47J25 | Iterative procedures involving nonlinear operators |
References:
| [1] | S. Akashi and W. Takahashi, Weak convergence theorem for an infinite family of demi-metric mappings in a Hilbert space, J. Nonlinear Convex Anal., 10 (2016), 2159-2169. · Zbl 1381.47043 |
| [2] | T.O. Alakoya and O.T. Mewomo, Viscosity S-Iteration Method with Inertial Technique and Self-Adaptive Step Size for Split Variational Inclusion, Equilibrium and Fixed Point Problems, Comput. Appl. Math., 41(1) (2021), 31 pp. · Zbl 1499.65251 |
| [3] | F. Alvarez and H. Attouch, An inertial proximal method for maximal monotone op-erators via discretization of a nonlinear oscillator with damping, Set-Valued Anal., 9 (2001), 3-11. · Zbl 0991.65056 |
| [4] | M.M. Alves and R.T. Marcavillaca, On inexact relative-error hybrid proximal extragra-dient, forward-backward and Tsengs modified forward-backward methods with inertial effects, Set-Valued Var. Anal., 28 (2020), 301-325. · Zbl 1445.90079 |
| [5] | A.S. Antipin, On a method for convex programs using a symmetrical modification of the Lagrange function, Ekonomika i Mat. Metody, 12 (1976), 1164-1173. · Zbl 0368.90115 |
| [6] | M. Bux, S. Ullah, M.S. Arif and K. Abodayeh, A self-Adaptive Technique for Solving Variational Inequalities: A New Approach to the Problem, J. Funct. Spaces, 3 (2022), 1-5. · Zbl 1508.47108 |
| [7] | L.C. Ceng, N. Hadjisavvas and N.C. Wong, Strong convergence theorem by a hybrid ex-tragradient like approximation method for variational inequalities and fixed point prob-lems, J. Glob. Optim., 46 (2010), 635-646. · Zbl 1198.47081 |
| [8] | Y. Censor, A. Gibali and S. Reich, Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space, Optim. Meth. Softw., 26 (2011), 827-845. · Zbl 1232.58008 |
| [9] | Y. Censor, A. Gibali and S. Reich, The subgradient extragradient method for solving variational inequalities in Hilbert space, J. Optim. Theory Appl., 148 (2011), 318-335. · Zbl 1229.58018 |
| [10] | Y. Censor, A. Gibali and S. Reich, Algorithms for the split variational inequality problem, Numer. Algorithms, 56 (2012), 301-323. · Zbl 1239.65041 |
| [11] | Y. Censor, A. Gibali and S. Reich, Extensions of Korpelevichs extragradient method for the variational inequality problem in Euclidean space, Optimization, 61 (2012), 1119-1132. · Zbl 1260.65056 |
| [12] | J. Chen, S. Liu and X. Chang, Modified Tsengs extragradient methods for variational inequality on Hadamard manifolds, Appl. Anal., 100 (2021), 2627-2640. · Zbl 07410391 |
| [13] | S. Dafermos, Traffic equilibrium and variational inequalities, Transp. Sci., 14 (1980), 42-54. |
| [14] | S.V Denisov, V.V. Semenov and L.M. Chabak, Convergence of the modified extragradient method for variational inequalities with non-Lipschitz operators, Cybern. Syst. Anal., 51 (2015), 757-765. · Zbl 1331.49010 |
| [15] | Q. Dong, Y. Cho, L. Zhong and T.M. Rassias, Inertial projection and contraction algo-rithms for variational inequalities, J. Glob. Optim., 70 (2018), 687-704. · Zbl 1390.90568 |
| [16] | W.M. Dyab, A.A Sakr, M.S Ibrahim and K. Wu, Variational Analysis of a Dually Polarized Waveguide Skew Loaded by Dielectric Slab, IEEE Microw. Wirel. Components Lett., 30 (2020), 737-740. |
| [17] | G. Fichera, Problemi elastostatici con vincoli unilaterali: il problema di Signorini con ambigue condizioni al contorno, Atti Accad. Naz. Lincci, Cl. Sci. Fis. Mat. Nat., Sez., 7 (1964), 91-140. · Zbl 0146.21204 |
| [18] | G. Fichera, Sul pproblem elastostatico di signorini con ambigue condizioni al contorno, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur, 34 (1963), 138-142. · Zbl 0128.18305 |
| [19] | E.C. Godwin, T.O. Alakoya, O.T. Mewomo and J.C. Yao, Relaxed inertial Tseng extra-gradient method for variational inequality and fixed point problems, Appl. Anal., (2022). DOI:10.1080/00036811.2022.2107913. · Zbl 07744477 · doi:10.1080/00036811.2022.2107913 |
| [20] | N. Hadjisavvas and S. Schaible, Quasimonotone variational inequalities in Banach spaces, J. Optimiz. Theory App., 90 (1996), 95-111. · Zbl 0904.49005 |
| [21] | B.S. He and L.Z. Liao, Improvements of some projection methods for monotone nonlin-ear variational inequalities, J. Optim. Theory Appl., 112 (2002), 111-128. · Zbl 1025.65036 |
| [22] | D.V. Hieu, P.K. Anh and L.D. Muu, Modified hybrid projection methods for finding common solutions to variational inequality problems, Comput. Optim. Appl., 66 (2017), 75-96. · Zbl 1368.65103 |
| [23] | D.V. Hieu and D.V. Thong, New extragradient-like algorithms for strongly pseudomono-tone variational inequalities, J. Glob. Optim., 70 (2018), 385-399. · Zbl 1384.65041 |
| [24] | H. Iiduka and W. Takahashi, Strong convergence theorems for nonexpansive mappings and inverse strongly monotone mappings, Nonlinear Anal., 61 (2005), 341-350. · Zbl 1093.47058 |
| [25] | C. Izuchukwu, A.A. Mebawondu, K.O Aremu, H.A. Abass and O.T. Mewomo, Viscos-ity iterative techniques for approximating a common zero of monotone operators in an Hadamard space, Rend. Circ. Mat. Palermo, (2019), https://doi.org/10.1007/s12215-019-00415-2. · Zbl 1461.47033 · doi:10.1007/s12215-019-00415-2 |
| [26] | C. Izuchukwu, G.N. Ogwo, A.A. Mebawondu and O.T. Mewomo, On finite family of monotone variational inclusion problems in reflexive Banach space, Politeh. Univ. Buchar. Sci. Bull. Ser. A Appl. Math. Phys., 82(3) (2020) 89-124. · Zbl 1505.47083 |
| [27] | P.D. Khanh and P.T. Vuong, Modified projection method for strongly pseudomonotone variational inequalities, J. Global Optim., 58 (2014), 341-350. · Zbl 1454.47095 |
| [28] | G.M. Korpelevich, The extragradient method for finding saddle points and other prob-lems, Ekonomikai Matematicheskie Metody, 12 (1976), 747-756. · Zbl 0342.90044 |
| [29] | H. Liu and J. Yang, Weak convergence of iterative methods for solving quasimonotone variational inequalities, Comput. Optim. Appl., 77 (2020), 491-508. · Zbl 07342388 |
| [30] | Y.V. Malitsky, Projected reflected gradient methods for monotone variational inequali-ties, SIAM J. Optim., 25 (2015), 502-520. · Zbl 1314.47099 |
| [31] | Y.V. Malitsky and V.V. Semenov, A hybrid method without extrapolation step for solving variational inequality problems, J. Glob. Optim., 61 (2015), 193-202. · Zbl 1366.47018 |
| [32] | M.A Noor and S. Ullah, Predictor-corrector self-adaptive methods for variational in-equalities, Transylv. Rev., 16 (2017), 4147-4152. |
| [33] | A.E. Ofem, A.A. Mebawondu, C. Agbonkhese, G.C. Ugwunnadi and O.K. Narain, Al-ternated inertial relaxed Tseng method for solving fixed point and quasi-monotone vari-ational inequality problems, Nonlinear Funct. Anal. Appl., 29 (1) ( 2024), 131-164. · Zbl 07868036 |
| [34] | D.O. Peter, A.A. Mebawondu, G.C. Ugwunndi, P. Pillay and O.K. Narain, Solving Quasimonotone Split Variational Inequality Problem and Fixed Point Problem In Hilbert Spaces, Nonlinear Funct. Anal. Appl., 28 (1) ( 2023), 205-235 · Zbl 1519.47091 |
| [35] | B.T Polyak, Some methods of speeding up the convergence of iteration methods, USSR Comput. Math. Math. Phys., 4(5) (1964), 1-17. · Zbl 0147.35301 |
| [36] | L.D. Popov, A modication of the ArrowHurwicz method for searching for saddle points, Mat. Zametki, 28 (1980), 777784. |
| [37] | S. Saejung and P. Yotkaew, Approximation of zeros of inverse strongly monotone oper-ators in Banach spaces, Nonlinear Anal., 75 (2012), 742-750. · Zbl 1402.49011 |
| [38] | D.R. Sahu and A.K. Singh, Inertial normal S-type Tsengs extragradient algorithm for solution of variational inequality problems, RAIRO Oper. Res., 55 (2021), 21652180. |
| [39] | S. Salahuddin, The extragradient method for quasi-monotone variational inequalities, Optimization, (2020), https://doi.org/10.1080/02331934.2020.1860979. · Zbl 1503.49015 · doi:10.1080/02331934.2020.1860979 |
| [40] | G. Stampacchia, Formes bilineaires coercitives sur les ensembles convexes, C. R. Math. Acad. Sci., 258 (1964), 4413-4416. · Zbl 0124.06401 |
| [41] | W. Takahashi, Introduction to Nonlinear and Convex Analysis, Yokohama Publishers, Yokohama, 2009. · Zbl 1183.46001 |
| [42] | W. Takahashi and M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone Mappings, J. Optim. Theory Appl., 118 (2003), 417-428. · Zbl 1055.47052 |
| [43] | D.V. Thong and D.V. Hieu, Modified Tsengs extragradient algorithms for variational inequality problems, J. Fixed Point Theory Appl., 79 (2018) 597-610. · Zbl 06945630 |
| [44] | D.V. Thong and D.V. Hieu, Some extragradient-viscosity algorithms for solving varia-tional inequality problems and fixed point problems, Numer. Algor., 82 (2019), 761-789. · Zbl 1441.47079 |
| [45] | D.V. Thong and P.T. Vuong, Modified Tsengs extragradient methods for solving pseudo-monotone variational inequalities, Optimization, 68 (2019), 22032222. |
| [46] | P. Tseng, A modified forward-backward splitting method for maximal monotone map-pings, SIAM J Control Optim., 38 (2000), 431-446. · Zbl 0997.90062 |
| [47] | F. Wang and H.K. Xu, Weak and strong convergence theorems for variational inequality and fixed point problems with Tsengs extragradient method, Taiwan. J. Math., 16 (2012), 1125-1136. · Zbl 1515.47107 |
| [48] | Y. Yao, A.M. Noor, I.K Noor, Y-C. Liou and H. Yaqoob, Modified extragradient methods for a system of variational inequalities in Banach spaces, Acta Appl. Math., 110(3) (2010), 1211-1224. · Zbl 1192.47065 |
| [49] | T.C. Yin, Y.K. Wu and C.F. Wen, An Iterative Algorithm for Solving Fixed Point Prob-lems and Quasimonotone Variational Inequalities, J. Math., 2022, Article ID 8644675, 9 pages, https://doi.org/10.1155/2022/8644675. · doi:10.1155/2022/8644675 |
| [50] | L. Zheng, A double projection algorithm for quasimonotone variational inequalities in Banach spaces, Inequal. Appl., 123 (2028), 1-20. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
