Convergence of Two-Stage Method with Bregman Divergence for Solving Variational Inequalities*

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

“…We proved weak and strong convergence results for approximating solutions of VI in real reflexive Banach spaces and provide numerical examples to illustrate the performance of our algorithms. Our results improve and extend the corresponding results of [7,12,18,34,35] in the literature.…”

“…(iii) Moreover, our algorithms does not require any prior estimate of the Lipschitz constant for their convergence. This improves the corresponding results of [7,8,12,18,35] in the literature. (iv) The strong convergence theorem proved in this paper is more desirable for solving optimization problems.…”

“…Remark 3.10 (i) We note that our results extend the results of [12,35] from finite-dimensional spaces to real Banach spaces. (ii) We also extend the result of Gibali [18] to solving pseudo-monotone variational inequalities and real Banach spaces.…”

“…Remark 3.2 We note that our Algorithm 3.1 is proposed in real Banach spaces while that of Nomirovski et al [34] and Denisov et al [12] were proposed in finite-dimensional spaces. Furthermore, our method is more general than that of [12,18,34,35] which used a fixed stepsize for all iterates. We will see in the following result that the stepsize rule defined by (3.2) is well defined.…”

“…Recently, Nomirovskii et al [35] proposed the following two-stage method using the Bregman divergence with operator f : H → R ∪ {+∞} being continuously differentiable and σ -strongly convex. Algorithm 1.1 Choose x 0 , y 0 ∈ C and positive number λ.…”