New projection methods with inertial steps for variational inequalities

Abstract

The inertial projection and contraction method and the inertial forward-backward-forward method have been studied for solving variational inequality problems due to the ability of these methods to work well with only one projection per iteration during computations. The inertial factor in these methods is chosen to be less than 1 in most of the papers that studied these methods. It is known from a computational point of view that the efficiency of these methods improves as the inertial factor approaches 1. In this paper, we modify both the inertial projection and contraction method and the inertial forward-backward-forward method with the possibility of inertial factor taken as 1. Our proposed methods also involve a step-size rule which does not depend on the Lipschitz constant of the cost function and without any line search rule. We analyse the weak convergence of the methods under appropriate conditions. We also modify the proposed methods so that strong convergence is obtained. Preliminary computational results show that our proposed methods are promising and show better performance than some variants of inertial projection and contraction methods and inertial forward-backward-forward methods where the inertial factor is assumed to be less than 1.

Keywords:

• Inertial step
• projection methods
• variational inequality
• weak and strong convergence

2010 MSC classifications:

• 47H05
• 47J20
• 47J25
• 65K15
• 90C25

Acknowledgements

The authors are grateful to the anonymous referees and the Handling Editor for their insightful comments which have improved the earlier version of the manuscript greatly.

Disclosure statement

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