Strong Convergence of a Bregman Extragradient Method With Non‐Monotone and Non‐Lipschitz Line‐Search in Reflexive Banach Spaces

ABSTRACT In this paper, we introduce a new Bregman extragradient method with a golden ratio line‐search for solving variational inequality problems in reflexive Banach spaces. Under the assumption that the involved operator is non‐monotone and non‐Lipschitz continuous, we prove that the sequence generated by the proposed algorithm converges strongly to a solution of the variational inequality problem. Numerical examples are provided to demonstrate the effectiveness of our method. The results obtained in this paper extend and improve many recent results in the literature.