Strong convergence results for solving bilevel split variational inequality problem

Abstract

In this article, we consider the problem of approximating the solution of bilevel split variational inequality problem in real Hilbert spaces. The underlying operators in the lower level problem are quasimonotone and Lipschitz continuous. The proposed algorithm is a combination of the modified subgradient extragradient and modified Tseng's extragradient methods. Compared with the existing modified subgragadient extragradient methods for solving bilevel split variational inequality problem, our suggested method does not required computation of the projections onto two half-spaces, containing the feasibility sets. The step sizes employed in our algorithm do not need the prior knowledge of the norm of the bounded linear operator and the the Lipschitz constants of the underlying operators. We obtain the strong convergence results of the new method using some mild conditions on the control parameters. The proposed method involves double inertial terms which permits it to accelerate its convergence speed. To show the advantage and potential of our method over some existing methods, we present some numerical experiments. Our results extend, unify, generalize and improve many existing results in this direction.