Strong convergence of modified inertial extragradient methods for non-Lipschitz continuous variational inequalities and fixed point problems. (English) Zbl 07830985
Summary: In this paper, we present two modified algorithms for finding a common element of the solution set of a quasimonotone variational inequality and the fixed point set of demicontractive mapping. The main advantage of such iterative methods is that it does not require Lipschitz continuity and the mapping only needs to be quasimonotonicity. Therefore, our algorithms can be applied to more general variational inequalities. Several numerical experiments are provided to verify the preponderance and efficiency of the proposed algorithms.
MSC:
| 47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |
| 47H10 | Fixed-point theorems |
| 49J35 | Existence of solutions for minimax problems |
| 68W10 | Parallel algorithms in computer science |
| 65K15 | Numerical methods for variational inequalities and related problems |
Keywords:
extragradient algorithm; quasimonotone variational inequality; fixed point set; demicontractive mappingReferences:
| [1] | Agwu, IK; Ishtiaq, U.; Saleem, N., Equivalence of novel IH-implicit fixed point algorithms for a general class of contractive maps, AIMS Math, 8, 841-872 (2023) · doi:10.3934/math.2023041 |
| [2] | Aubin, JP; Ekeland, I., Applied nonlinear analysis (1984), New York: Wiley, New York · Zbl 0641.47066 |
| [3] | Baiocchi, C.; Capelo, A., Variational and quasivariational inequalities applications to free boundary problems (1984), New York: Wiley, New York · Zbl 0551.49007 |
| [4] | Ceng, LC; Yao, JC, Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems, Taiwan J Math, 10, 1293-1303 (2006) · Zbl 1110.49013 |
| [5] | Ceng LC, Petrusel A, Yao JC (2019a) Systems of variational inequalities with hierarchical variational inequality constraints for Lipschitzian pseudocontractions. Fixed Point Theory 20:113-133 · Zbl 1430.49004 |
| [6] | Ceng LC, Petruel A, Yao JC (2019b) On Mann viscosity subgradient extragradient algorithms for fixed point problems of finitely many strict pseudocontractions and variational inequalities. Mathematics 7:925 |
| [7] | Ceng, LC; Petrusel, A.; Qin, X., A Modified inertial subgradient extragradient method for solving pseudo-monotone variational inequalities and common fixed point problems, Fixed Point Theory, 21, 93-108 (2020) · Zbl 1477.47060 · doi:10.24193/fpt-ro.2020.1.07 |
| [8] | Censor, Y.; Gibali, A.; Reich, S., The subgradient extragradient method for solving variational inequalities in Hilbert space, J Optim Theory Appl, 148, 318-335 (2011) · Zbl 1229.58018 · doi:10.1007/s10957-010-9757-3 |
| [9] | Cottle, RW; Yao, JC, Pseudo-monotone complementarity problems in Hilbert space, J Optim Theory Appl, 75, 281-295 (1992) · Zbl 0795.90071 · doi:10.1007/BF00941468 |
| [10] | Denisov, SV; Semenov, VV; Chabak, LM, Convergence of the modified extragradient method for variational inequalities with non-Lipschitz operators, Cybern Syst Anal, 51, 757-765 (2015) · Zbl 1331.49010 · doi:10.1007/s10559-015-9768-z |
| [11] | Gibali, A.; Shehu, Y., An efficient iterative method for finding common fixed point and variational inequalities in Hilbert spaces, Optimization, 68, 13-32 (2019) · Zbl 07007063 · doi:10.1080/02331934.2018.1490417 |
| [12] | Godwin, EC; Mewomo, OT; Araka, NN; Okeke, GA; Ezeamama, GC, An inertial scheme for solving bi-level variational inequalities and the fixed point problem with pseudomonotone and \(\varrho \)-demimetric mappings, Appl Set-Valued Anal Optim, 4, 251-267 (2022) |
| [13] | Hu, X.; Wang, J., Solving pseudo-monotone variational inequalities and pseudo-convex optimization problems using the projection neural network, IEEE Trans Neural Netw, 17, 1487-1499 (2006) · doi:10.1109/TNN.2006.879774 |
| [14] | Iusem, A.; Otero, RG, Inexact versions of proximal point and augmented Lagrangian algorithms in Banach spaces, Numer Funct Anal Optim, 22, 609-640 (2001) · Zbl 1018.90067 · doi:10.1081/NFA-100105310 |
| [15] | Jabeen, S.; Noor, MA; Noor, KI, Inertial methods for solving system of quasi variational inequalities, J Adv Math Stud, 15, 1-10 (2022) · Zbl 1496.49008 |
| [16] | Karamardian, S., Complementarity problems over cones with monotone and pseudo-monotone maps, J Optim Theory Appl, 18, 445-454 (1976) · Zbl 0304.49026 · doi:10.1007/BF00932654 |
| [17] | Kinderlehrer, D.; Stampacchia, G., An introduction to variational inequalities and their applications (1980), New York: Academic Press, New York · Zbl 0457.35001 |
| [18] | Konnov, IV, Combined relaxation methods for variational inequalities (2001), Berlin: Springer, Berlin · Zbl 0982.49009 · doi:10.1007/978-3-642-56886-2 |
| [19] | Korpelevich, GM, The extragradient method for finding saddle points and other problems, Ekon Mat Metody, 12, 747-756 (1976) · Zbl 0342.90044 |
| [20] | Linh, HM; Reich, S.; Thong, DV, Analysis of two variants of an inertial projection algorithm for finding the minimum-norm solutions of variational inequality and fixed point problems, Numer Algorithms, 89, 1695-1721 (2022) · Zbl 1505.47084 · doi:10.1007/s11075-021-01169-8 |
| [21] | Okeke, GA; Abbas, M.; De la Sen, M., Accelerated modified tsengs extragradient method for solving variational inequality problems in Hilbert spaces, Axioms, 10, 4, 248 (2021) · doi:10.3390/axioms10040248 |
| [22] | Saejung, S.; Yotkaew, P., Approximation of zeros of inverse strongly monotone operators in Banach spaces, Nonlinear Anal, 75, 742-750 (2012) · Zbl 1402.49011 · doi:10.1016/j.na.2011.09.005 |
| [23] | Saleem, N.; Agwu, IK; Ishtiaq, U., Strong convergence theorems for a finite family of enriched strictly pseudocontractive mappings and \(\phi\) T-enriched Lipschitizian mappings using a new modified mixed-type Ishikawa iteration scheme with error, Symmetry, 14, 1032 (2022) · doi:10.3390/sym14051032 |
| [24] | Shehu, Y.; Ogbuisi, FU, An iterative algorithm for approximating a solution of split common fixed point problem for demi-contractive maps, Dyn Contin Discrete Impulse Syst Ser B Appl Algorithms, 23, 205-216 (2016) · Zbl 1347.47045 |
| [25] | Shehu, Y.; Cholamjiak, P., Iterative method with inertial for variational inequalities in Hilbert spaces, Calcolo, 56, 1-21 (2019) · Zbl 1482.47127 · doi:10.1007/s10092-018-0300-5 |
| [26] | Solodov, MV; Tseng, P., Modifified projection-type methods for monotone variational inequalities, SIAM J Control Optim, 34, 1814-1830 (1996) · Zbl 0866.49018 · doi:10.1137/S0363012994268655 |
| [27] | Tan, B.; Zhou, Z.; Li, S., Viscosity-type inertial extragradient algorithms for solving variational inequality problems and fixed point problems, J Appl Math Comput, 68, 1387-1411 (2022) · Zbl 07532886 · doi:10.1007/s12190-021-01576-z |
| [28] | Thong DV, Hieu DV (2018a) Weak and strong convergence theorems for variational inequality problems. Numer Algorithms 78:1045-1060 · Zbl 1398.65376 |
| [29] | Thong DV, Hieu DV (2018b) New extragradient methods for solving variational inequality problems and fixed point problems. Fixed Point Theory Appl 20:129 · Zbl 1490.65104 |
| [30] | Thong, DV; Cho, YJ, A strong convergence theorem for Tsengs extragradient method for solving variational inequality problems, Optim Lett, 14, 1157-1175 (2020) · Zbl 1445.49005 · doi:10.1007/s11590-019-01391-3 |
| [31] | Tseng, P., A modified forward-backward splitting method for maximal monotone mappings, SIAM J Control Optim, 38, 431-446 (2000) · Zbl 0997.90062 · doi:10.1137/S0363012998338806 |
| [32] | Wang, F.; Xu, HK, Weak and strong convergence theorems for variational inequality and fixed point problems with Tsengs extragradient method, Taiwan J Math, 16, 1125-1136 (2012) · Zbl 1515.47107 · doi:10.11650/twjm/1500406682 |
| [33] | Wang, M.; Ishtiaq, U.; Saleem, N., Approximating common solution of minimization problems involving asymptotically quasi-nonexpansive multivalued mappings, Symmetry, 14, 2062 (2022) · doi:10.3390/sym14102062 |
| [34] | Yao, Y.; Postolache, M., Iterative methods for pseudomonotone variational inequalities and fixed point problems, J Optim Theory Appl, 155, 273-287 (2012) · Zbl 1267.90158 · doi:10.1007/s10957-012-0055-0 |
| [35] | Yao, Y.; Shahzad, N.; Yao, JC, Convergence of Tseng-type self-adaptive algorithms for variational inequalities and fixed point problems, Carpathian J Math, 37, 541-550 (2021) · Zbl 1485.47111 · doi:10.37193/CJM.2021.03.15 |
| [36] | Yao, Y.; Iyiola, OS; Shehu, Y., Subgradient extragradient method with double inertial steps for variational inequalities, J Sci Comput, 90, 1-29 (2022) · Zbl 07458297 · doi:10.1007/s10915-021-01751-1 |
| [37] | Ye, M.; He, Y., A double projection method for solving variational inequalities without monotonicity, Comput Optim Appl, 60, 141-150 (2016) · Zbl 1308.90184 · doi:10.1007/s10589-014-9659-7 |
| [38] | Zhao, X.; Yao, Y., Modified extragradient algorithms for solving monotone variational inequalities and fixed point problems, Optimization, 69, 1987-2002 (2020) · Zbl 07249882 · doi:10.1080/02331934.2019.1711087 |
| [39] | Zhao, TY; Wang, DQ; Ceng, LC, Quasi-inertial Tsengas extragradient algorithms for pseudomonotone variational inequalities and fixed point problems of quasi-nonexpansive operators, Numer Funct Anal Optim, 42, 69-90 (2020) · Zbl 07336635 · doi:10.1080/01630563.2020.1867866 |
| [40] | Zhou, H.; Qin, X., Fixed points of nonlinear operators (2020), Berlin: De Gruyter, Berlin · Zbl 1440.47001 · doi:10.1515/9783110667097 |
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.
