A novel class of forward-backward explicit iterative algorithms using inertial techniques to solve variational inequality problems with quasi-monotone operators

Abstract

<abstract><p>The theory of variational inequalities is an important tool in physics, engineering, finance, and optimization theory. The projection algorithm and its variants are useful tools for determining the approximate solution to the variational inequality problem. This paper introduces three distinct extragradient algorithms for dealing with variational inequality problems involving quasi-monotone and semistrictly quasi-monotone operators in infinite-dimensional real Hilbert spaces. This problem is a general mathematical model that incorporates a set of applied mathematical models as an example, such as equilibrium models, optimization problems, fixed point problems, saddle point problems, and Nash equilibrium point problems. The proposed algorithms employ both fixed and variable stepsize rules that are iteratively transformed based on previous iterations. These algorithms are based on the fact that no prior knowledge of the Lipschitz constant or any line-search framework is required. To demonstrate the convergence of the proposed algorithms, some simple conditions are used. Numerous experiments have been conducted to highlight the numerical capabilities of algorithms.</p></abstract>

A self-adaptive iterative method for a split equality problem

“…However, in many applications, F(x) and G(y) may not be Lipschitz continuous (or it could be difficult to verify their Lipschitz continuity condition). Motivated by the results of Latif and Eslamian [30], Panyanak et al [21], and the ongoing research in this direction, in this paper, we introduce an iterative algorithm for the split equality problem with an equilibrium problem, a variational inequality problem, and a fixed point problem of nonexpansive semigroups. We establish a strong convergence theorem of solutions by the uniformly continuity rather than the Lipchitz continuity of these mappings.…”

Solving quasimonotone and non-monotone variational inequalities

“…In this study we are concerned about solving the VI (1.1) for the case when A is quasimonotone, i.e., when V I V D , (see Panyanak et al 2023;Wang et al 2023). Recently, Liu and Yang (2020) proposed a new self-adaptive method for solving the variational inequalities with quasimonotone operator (or without monotonicity).…”