Modified inertial-like subgradient extragradient rule with adaptive step sizes for hierarchical variational inequalities with systems of variational inequalities

Abstract

In this paper, based on the viscosity approximation method, hybrid steepest-descent method and Korpelevich extragradient method, we suggest and analyze two modified inertial-like subgradient extragradient algorithms with adaptive step sizes for solving a hierarchical variational inequality (HVI) with a variational inequality problem for pseudomonotone and Lipschitz continuous mapping (VIP), a general system of variational inequalities for two inverse-strongly monotone mappings (GSVI) and a common fixed-point problem of finitely many nonexpansive mappings and an asymptotically nonexpansive mapping (CFPP) constraints. Under mild restrictions, we demonstrate the strong convergence of the proposed algorithms to a common solution of the VIP, GSVI and CFPP, which is also the unique solution of this HVI.