Email this article A NEW ITERATIVE METHOD FOR SOLVING PSEUDOMONOTONE VARIATIONAL INEQUALITIES
About this article
Received: 16/02/23 Revised: 27/04/23 Published: 28/04/23Authors
1. Dang Hong Linh
, Hanoi University of Science and Technology2. Nguyen Tat Thang, Thai Nguyen University
Abstract
In this paper, we introduce a modified algorithm for pseudomono- tone variational inequalities. This problem has many important applications in different fields such as optimization problem, Nash equilibrium problem, game theory, traffic equilibrium problem, fixed point problem. The proposed algorithm bases on the self-adaptive method and the modified Popov extragradient method that have been applied to solve many other problems with Lipschitz continuous mapping. The advantage of the algorithm is that it only needs to compute one value of the inequality mapping as well as it does not require knowing the Lipschitz constants of the variational inequality mapping. Moreover, our algorithm does not require its step-sizes tending to zero. This feature helps to speed up our method. The convergence of the method has been proved based on the specified conditions of the parameters. A numerical experiment in Euclidean spaces is given to illustrate the convergence of the new algorithm.
Keywords
Full Text:
PDFReferences
[1] H. Iiduka, “Fixed point optimization algorithm and its application to power control in CDMA data networks,” Mathematical Programming, vol. 133, pp. 227–242, 2012.
[2] H. Iiduka and I. Yamada, “An ergodic algorithm for the power-control games for CDMA data networks,” Journal of Mathematical Modelling and Algorithms, vol. 8, pp. 1–18, 2009.
[3] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980.
[4] G.M. Korpelevich, “The extragradient method for finding saddle points and other problems,” Ekonomikai Matematicheskie Metody, vol. 12, pp. 747–756, 1976.
[5] P.K. Anh and T.N. Hai, “Splitting extragradient-like algorithms for strongly pseudomonotone equilibrium problems,” Numerical Algorithms, vol. 76, pp. 67–91, 2017.
[6] D.V. Hieu and D.V. Thong, “New extragradient-like algorithms for strongly pseudomonotone variational inequalities,” Journal of Global Optimization, vol. 70, pp. 385–399, 2018.
[7] J. Yang and H. Liu, “Strong convergence result for solving monotone variational inequalities in Hilbert space,” Numerical Algorithms, vol. 80, pp. 741–752, 2019.
[8] L.D. Popov, “A modification of the Arrow-Hurwicz method for searching for saddle points,” Mathemati-cal notes of the Academy of Sciences of the USSR, vol. 28, no. 5, pp. 777–784, 1980.
[9] Y.V. Malitsky and V.V. Semenov, “An extragradient algorithm for monotone variational inequalities,” Cybernetics and Systems Analysis, vol. 50, pp. 271–277, 2014.
[10] T.N. Hai, “Two modified extragradient algorithms for solving variational inequalities,” Journal of Global Optimization, vol. 78, no. 1, pp. 91-106, 2020.
[11] H.H. Bauschke and P.L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, New York, 2011.
[12] R.P. Agarwal, D. O’Regan, and D.R. Sahu, Fixed Point Theory for Lipschitzian-type Mappings with Applications, Springer, 2009.
[13] H.H. Bauschke and J.M. Borwein, “On projection algorithms for solving convex feasibility problems,” SIAM Review, vol. 38, pp. 367–426, 1996.
DOI: https://doi.org/10.34238/tnu-jst.7344
Refbacks
- There are currently no refbacks.