A Two-Step Inertial Bregman-Tseng Method for Non-Monotone Variational Inequality Problems in Reflexive Banach Spaces

Abstract

In this paper, we introduce a novel two-step inertial Bregman--Tseng algorithm for solving non-monotone variational inequality problems in reflexive Banach spaces. The proposed method combines a two-step inertial extrapolation scheme with a Bregman forward-backward-forward (Tseng-type) framework, leading to a computationally efficient iterative process that avoids auxiliary projections onto complex sets. Without imposing monotonicity assumptions on the cost operator, we establish strong convergence of the proposed algorithm under standard conditions, including Lipschitz continuity and mild control requirements on the inertial and relaxation parameters. The convergence analysis is carried out using a self-adaptive line search rule and relies on the assumption that the solution set of the corresponding dual variational inequality is nonempty. Furthermore, we provide numerical experiments to demonstrate the effectiveness of the proposed method and highlight its advantages in terms of computational simplicity and efficiency when compared with existing projection-type and extragradient methods. The results presented in this work extend and complement several important contributions in the current literature on variational inequality problems in Banach spaces.