New extragradient-like algorithms for strongly pseudomonotone variational inequalities
- 269 Downloads
The paper considers two extragradient-like algorithms for solving variational inequality problems involving strongly pseudomonotone and Lipschitz continuous operators in Hilbert spaces. The projection method is used to design the algorithms which can be computed more easily than the regularized method. The construction of solution approximations and the proof of convergence of the algorithms are performed without the prior knowledge of the modulus of strong pseudomonotonicity and the Lipschitz constant of the cost operator. Instead of that, the algorithms use variable stepsize sequences which are diminishing and non-summable. The numerical behaviors of the proposed algorithms on a test problem are illustrated and compared with those of several previously known algorithms.
KeywordsVariational inequality problem Monotone operator Pseudomonotone operator Strongly monotone operator Strongly pseudomonotone operator Extragradient method Subgradient extragradient method Projection method
Mathematics Subject Classification65Y05 65K15 68W10 47H05 47H10
The authors would like to thank the Associate Editor and two anonymous referees for their valuable comments and suggestions which helped us very much in improving the original version of this paper. The first author was partially supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 101.01-2017.315. The second author was partially funded by NAFOSTED under Grant No. 101.02-2017.15 and by Vietnam Institute for Advanced Study in Mathematics (VIASM).
- 6.Dong, Q.L., Cho, Y.J., Zhong, L.L., Rassias, T.M.: Inertial projection and contraction algorithms for variational inequalities. J. Glob. Optim. (2017). doi: 10.1007/s10898-017-0506-0
- 11.Hieu, D.V.: An explicit parallel algorithm for variational inequalities. Bull. Malays. Math. Sci. Soc. (2017). doi: 10.1007/s40840-017-0474-z
- 14.Hieu, D.V.: An extension of hybrid method without extrapolation step to equilibrium problems. J. Ind. Manag. Optim. (2016). doi: 10.3934/jimo.2017015
- 15.Hieu, D.V.: New extragradient method for a class of equilibrium problems in Hilbert spaces. Appl. Anal. (2017). doi: 10.1080/00036811.2017.1292350
- 18.Khanh, P.D.: A new extragradient method for strongly pseudomonotone variational inequalities. Numer. Funct. Anal. Optim. 37(9), 1131–1143 (2016). doi: 10.1080/01630563.2016.1212372
- 32.Thong, D.V., Hieu, D.V.: An inertial method for solving split common fixed point problems. J. Fixed Point Theory Appl. (2017). doi: 10.1007/s11784-017-0464-7
- 33.Thong, D.V.: Viscosity approximation methods for solving fixed point problems and split common fixed point problems. J. Fixed Point Theory Appl. 19(2), 1481–1499 (2017). doi: 10.1007/s11784-016-0323-y
- 36.Verma, R.U.: General system of strongly pseudomonotone nonlinear variational inequalities based on projection systems. J. Inequal. Pure Appl. Math. 8(1–9) (2007)Google Scholar