A new self-adaptive algorithm for solving pseudomonotone variational inequality problems in Hilbert spaces

ABSTRACT

In this paper, we revisit the subgradient extragradient method for solving a pseudomonotone variational inequality problem with the Lipschitz condition in real Hilbert spaces. A new algorithm based on the subgradient extragradient method with the technique of choosing a new step size is proposed. The weak convergence of the proposed algorithm is established under the pseudomonotonicity and the Lipschitz continuity as well as without using the sequentially weakly continuity of the variational inequality mapping and the nonasymptotic $O\left(1/n\right)$ convergence rate of the proposed algorithm is presented, while the strong convergence theorem of the proposed algorithm is also proved under the strong pseudomonotonicity and the Lipschitz continuity hypotheses. In order to show the computational effectiveness of our algorithm, some numerical results are provided.

Keywords:

• Subgradient extragradient method
• inertial method
• variational inequality problem
• pseudomonotone mapping
• Lipschitz continuity
• convergence rate

AMS Classifications:

• 65Y05
• 65K15
• 68W10
• 47H05
• 47H10

Acknowledgements

The authors would like to thank the Editor and the two referees for their valuable comments and suggestions which helped us very much in improving and presenting the original version of this paper. This research is funded by National Economics University, Hanoi, Vietnam.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research is funded by National Economics University, Hanoi, Vietnam.