Weak and strong convergence Bregman extragradient schemes for solving pseudo-monotone and non-Lipschitz variational inequalities

In this paper, we introduce Bregman subgradient extragradient methods for solving variational inequalities with a pseudo-monotone operator which are not necessarily Lipschitz continuous. Our algorithms are constructed such that the stepsizes are determined by an Armijo line search technique, which improves the convergence of the algorithms without prior knowledge of any Lipschitz constant. We prove weak and strong convergence results for approximating solutions of the variational inequalities in real reflexive Banach spaces. Finally, we provide some numerical examples to illustrate the performance of our algorithms to related algorithms in the literature.