Accelerated projection methods for quasimonotone bilevel variational inequality problems with applications

Abstract

In this paper, we propose two novel alternated multi-step inertial projection algorithms with self-adaptive step sizes. These algorithms are employed to solve the quasimonotone bilevel variational inequality problem (QBVIP, where VIP denotes a variational inequality problem) with a variational inclusion constraint in a real Hilbert space, where the QBVIP involves a strongly monotone mapping at the upper-level VIP and quasimonotone mapping at the lower-level. We establish the strong convergence of the proposed algorithms under some suitable conditions. Furthermore, we demonstrate the applicability of these algorithms to the split feasibility problem and the generalized Nash equilibrium problem. These works extend and develop some of the existing results in the literature. Finally, we apply our results to LASSO problem and numerical experiments demonstrate the effectiveness and superiority of the proposed algorithms.