Analysis of two versions of relaxed inertial algorithms with Bregman divergences for solving variational inequalities

Abstract

In this paper, we introduce and analyze two new inertial-like algorithms with the Bregman divergences for solving the pseudomonotone variational inequality problem in a real Hilbert space. The first algorithm is inspired by the Halpern-type iteration and the subgradient extragradient method and the second algorithm is inspired by the Halpern-type iteration and Tseng's extragradient method. Under suitable conditions, we prove some strong convergence theorems of the proposed algorithms without assuming the Lipschitz continuity and the sequential weak continuity of the given mapping. Finally, we give some numerical experiments with various types of Bregman divergence to illustrate the main results. In fact, the results presented in this paper improve and generalize the related works in the literature.