An efficient inertial type iterative algorithm to approximate the solutions of quasi variational inequalities in real Hilbert spaces. (English) Zbl 1493.47090
Summary: In this article, we design a projection type iterative algorithm with two inertial steps for solving quasi-variational inequalities with Lipschitz continuous and strongly monotone mappings in real Hilbert spaces. We establish different strong convergence results through this algorithm. We give a non-trivial example to validate one of our results and to illustrate the efficiency of the proposed algorithm compared with an already existing one. We also present some numerical experiments to demonstrate the potential applicability and computing performance of our algorithm compared with some other algorithms existing in the literature. The results obtained herein are generalizations and substantial improvements of some earlier results.
MSC:
| 47J25 | Iterative procedures involving nonlinear operators |
| 47H05 | Monotone operators and generalizations |
| 65K15 | Numerical methods for variational inequalities and related problems |
Keywords:
quasi-variational inequalities; inertial projection-type method; strong monotonicity; Lipschitz continuous; Hilbert spaces; strong convergenceSoftware:
QVILIBReferences:
| [1] | Ansari, Q.H. (ed.): Nonlinear Analysis: Approximation Theory. Optimization and Applications, Springer, Berlin (2014) |
| [2] | Beck, A.; Teboulle, M., A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci., 2, 1, 183-202 (2009) · Zbl 1175.94009 · doi:10.1137/080716542 |
| [3] | Facchinei, F.; Kanzow, C.; Sagratella, S., QVILIB: a library of quasi-variational inequality test problems, Pac. J. Optim., 9, 225-250 (2013) · Zbl 1267.65078 |
| [4] | Liu, Q., A convergence theorem of the sequence of Ishikawa iterates for quasi-contractive mappings, J. Math. Anal. Appl., 146, 2, 301-305 (1990) · Zbl 0392.93028 · doi:10.1016/0022-247X(77)90110-X |
| [5] | Mijajlović, N.; Jaćimović, M.; Noor, MA, Gradient-type projection methods for quasi-variational inequalities, Optim. Lett., 13, 1885-1896 (2019) · Zbl 1433.90171 · doi:10.1007/s11590-018-1323-1 |
| [6] | Noor, MA; Oettli, W., On general nonlinear complementarity problems and quasi equilibria, Le Mathematiche, 49, 313-331 (1994) · Zbl 0839.90124 |
| [7] | Shehu, Y.; Gibali, A.; Sagratella, S., Inertial projection-type methods for solving quasi-variational inequalities in real Hilbert spaces, J. Optim. Theory Appl., 184, 877-894 (2020) · Zbl 1513.47131 · doi:10.1007/s10957-019-01616-6 |
| [8] | Xu, HK, Iterative algorithms for nonlinear operators, J. Lond. Math. Soc., 66, 240-256 (2002) · Zbl 1013.47032 · doi:10.1112/S0024610702003332 |
| [9] | Zegeye, H., Shahzad, N.: Convergence of Mann’s type iteration method for generalized asymptotically nonexpansive mappings. Comput. Math. Appl. 62(11), 4007-4014 (2011) · Zbl 1471.65043 |
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.
