A unified framework for three accelerated extragradient methods and further acceleration for variational inequality problems

Abstract

The main strategy of this paper is intended to speed up the convergence of the inertial Mann iterative method and further speed up it through the normal S-iterative method for a certain class of nonexpansive type operators that are linked with variational inequality problems. Our new convergence theory permits us to settle down the difficulty of unification of Korpelevich's extragradient method, Tseng's extragardient method, and subgardient extragardient method for solving variational inequality problems through an auxiliary algorithmic operator, which is associated with seed operator. The paper establishes an interesting fact that the relaxed inertial normal S-iterative extragradient methods do influence much more on convergence behaviour. Finally, the numerical experiments are carried out to illustrate that the relaxed inertial iterative methods, in particular the relaxed inertial normal S-iterative extragradient methods, may have a number of advantages over other methods in computing solutions of variational inequality problems in many cases.