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

Abstract

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.

Bregman subgradient extragradient method with monotone self-adjustment stepsize for solving pseudo-monotone variational inequalities and fixed point problems

“…(iii) We proved a strong convergence theorem for finding common solution of variational inequalities and fixed point of Bregman quasi-nonexpansive mapping using Bregman technique. This improves and extends the results of [1,9,29,25,28].…”

Convergence analysis for variational inequalities and fixed point problems in reflexive Banach spaces

“…More so, the stepsize is determined by a line search process which is computationally expensive. Furthermore, Jolaoso and Aphane [33] introduced a Bregman subgradient extragradient method with a line search technique for solving variational inequalities in a real reflexive Banach space. Very recently, Jolaoso and Shehu [34] introduced a single Bregman projection method with self-adaptive stepsize selection technique for solving variational inequalities in a real reflexive Banach space.…”

“…70n+3 , μ = 0.26, α 0 = 0.0025, u n = x 0 n+1 . We compare the performance of Algorithm 1 with Algorithm 3.6 of [33] (namely BSEM) and Algorithm 3.3 of [34] using the following initial value:…”

A Generalized Viscosity Inertial Projection and Contraction Method for Pseudomonotone Variational Inequality and Fixed Point Problems

“…The solution set of (1) is denoted by S. The VIP is a powerful tool for studying many nonlinear problems arising in mechanics, optimization, control network, equilibrium problems, and so forth; see References [1][2][3]. Due to this importance, the problem has drawn the attention of many researchers who had studied its existence of solution and proposed various iterative methods such as the extragradient method [4][5][6][7][8][9], subgradient extragradient method [10][11][12][13][14], projection and contraction method [15,16], Tseng's extragradient method [17,18] and Bregman projection method [19,20] for approximating its solution in various dimensions. The operator A : Ω → H is said to be 1. β-strongly monotone on Ω if there exists β > 0 such that Ax − Ay, x − y ≥ β x − y ∀x, y ∈ Ω;…”

“…It is easy to see from (1) ⇒ (2) ⇒ (4) and ( 1) ⇒ (3) ⇒ (4), but the converse implications do not hold in general; see, for instance Reference [16,19].…”