Weak and strong convergence of a modified adaptive generalized Popov's algorithm for solving variational inequality problems

Abstract

In this paper, we introduce a modified adaptive step-size algorithm by basing a generalized Popov's method to solve variational inequality problems in a real Hilbert space. The algorithm only needs to act on the operator once time in each iteration, which greatly reduces the calculation time and calculation difficulty. Of particular interesting, by using our new same generalized Popov's method, we also weaken the conditions of the operator about weak continuity and quasi-monotone property. Of particular interesting, by using our new same generalized Popov's method, we can obtain a weak convergence result under the condition that the operator A is quasi-monotone and a R-linear convergence result under the condition that the operator A is strongly pseudo-monotone in the variational inequality, respectively. Finally, the feasibility and effectiveness of the algorithm are verified by some numerical experiments.