This paper presents two inertial viscosity Mann-type extrapolated algorithms for finding a common solution to the variational inequality problem involving a monotone and Lipschitz continuous operator and the fixed-point problem for a demicontractive mapping in real Hilbert spaces. The proposed algorithms feature an adaptive step size strategy, computed iteratively, which circumvents the need for prior knowledge of the operator’s Lipschitz constant. Under appropriate assumptions, we establish two strong convergence theorems guaranteeing the robustness of the methods. Furthermore, we provide a comparative performance analysis of the proposed algorithms against some existing strongly convergent schemes, supported by numerical experiments with MATLAB-based graphical illustrations.

Abstract In this paper, we present and study two strongly convergent two-step inertial accelerated algorithms for solving pseudo-monotone variational inequality problem. Our methods are modification and extension of the Halpern-type method previously studied in the literature. Unique feature of our methods is their ability to handle non-Lipschitz continuous and pseudo-monotone operator through a novel line-search rule. Additionally, the two-step inertial technique incorporated in our methods account for the improvement in the rate of convergence. Finally, we compare our methods with some well-known methods in the literature through numerical experiments. The numerical results reveal that the proposed methods perform better than these methods.

Approximate subgradient extragradient methods for solving variational inequality problems: Convergence analysis and applications in signal and image processing

A novel iterative algorithm for solving a class of variational inequality problems: revisions to ‘On the intermixed method for mixed variational inequality problems: another look and some corrections’

A Stochastic Double Inertial Method for Solving Stochastic Variational Inequality Problem

In this paper, we design some generalized split problems which can be seen as an extended form of the split variational inequality problems. We present several iterative algorithms for solving generalized split problems and demonstrate the weak convergence results under some appropriate assumptions within the context of real Hilbert spaces. Finally, we support these results with the help of numerical examples in both the finite and infinite dimensional spaces. As a result of this work, a new direction will be opened in studying split problems.

In 2003 W. Takahashi and M. Toyoda showed the weak convergence of an iteration process of finding the solution of a variational inequality problem for an inverse strongly monotone mapping. In the present paper, we show that for the same process, most of its iterates are approximate common solutions for a finite family of variational inequalities induced by inverse strongly monotone mappings.

Convergence and stability analysis of a set-valued mixed variational inequality problem via three-step iterative approximation scheme

This paper presents an enhanced inertial Tseng’s extragradient method designed to address variational inequality problems involving pseudomonotone operators, along with fixed point problems governed by quasi-nonexpansive operators in real Hilbert spaces. Provided that the parameters satisfy appropriate conditions, the proposed method is shown to converge strongly. Finally, we provide computational results and illustrate their utility through optimal control applications. These aim to show the efficacy and superiority of the proposed algorithm compared with some existing algorithms.

A novel algorithm for strongly monotone variational inequalities with the multiple-sets split variational inequality problem constraints

ABSTRACT We investigate the split mixed variational inequality problem with multiple output sets in real Hilbert spaces. By incorporating multiple inertial steps, utilizing a general index control mapping, and employing either hybrid methods or shrinking projection techniques, we introduce two novel cyclic projection algorithms for solving this problem without requiring the inverse strong monotonicity assumption on the associated operators. In addition, the control parameters are designed to be self‐adaptive.

Halpern-type Bregman Projection Algorithms for Split Variational Inequality Problems

We extend the notion of asymptotically nonexpansive mapping to the more general class, namely, e-enriched asymptotically nonexpansive mappings. It is shown, with an example, that the class of e-enriched asymptoticallynonexpansive mappings is more general than the class of asymptotically nonexpansive mappings. Certain weak and strong convergence theorems are then proved for the iterative approximation of split common fixed point problem involving the class of(e,ϑ)-enriched strictly quasi-pseudocontractive mappings and the class of e-enriched asymptotically nonexpansive mappings in the domain of two Banach spaces. Furthermore, a significant result for the hierarchical variational inequality problem is obtained as a consequence of our main result.

The basic purpose of this article is to introduce a generalized version of pseudo-monotone variational inequality in the setting of complete $\mathcal{CAT}(0)$ spaces and to present some strong and $\Delta$-convergence results for the existence of solutions for the respective variational inequality problem. Algorithm 1 and 2 are proposed in accordance with pseudo-monotone and $\alpha$-strongly pseudo-monotone mappings to prove our results under some conditions. A numerical implication of our proposed algorithm is also presented.

Analysis of Tseng algorithm with inertial extrapolation step for stochastic variational inequality problem

Inertial forward-reflected-backward method for solving bilevel variational inequality problem

Accelerated projection methods for quasimonotone bilevel variational inequality problems with applications

This study aims at extending the idea of common solutions to problems in classical functional analysis to accommodate situations where there are randomness in the system, as real life problems are, mostly, of this nature. A common solution to random split feasibility and random variational inequality problems, called random split variational inequality problem, is sought through fixed point theory, using a nonexpansive operator. A random type of the two-step Wang’s algorithm is used to obtain a unique solution to the problem; and a strong convergence to this unique solution is proven. The result is applied to optimal tax policy problem and is seen to be adequate in solving the problem, yielding tax rates of 14.79% and 13.91% for the two categories of businesses. This result extends, and unifies some established results in the literature on deterministic functional analysis.

In this paper, we have presented a novel inertial-viscosity approximation technique for finding the common element (CE) of solution set of two variational inequality problems in the real Hilbert spaces. First, we investigate the CE of the set of fixed points (FPs) of two Nonexpansive (NE) mappings and then apply our result to find the CE of the solution set of two variational inequality problems. We also discussed on the strong convergence of the proposed algorithm under some moderate circumstances. Our result improves, extends and generalizes some published results in this direction. Furthermore, our results are supported by some numerical examples.

In this work, we explore a Maxwell variational inequality arising from Bean’s critical-state model in type-II superconductivity. Instead of the coupled electric and magnetic fields, we study an equivalent model based on the magnetic vector potential. The well-posedness of this transformed Maxwell variational inequality comes naturally from the hyperbolic Maxwell (quasi-)variational inequality arguments. We establish error estimates for both the spatially semi-discrete scheme and the fully discrete scheme. We propose two kinds of methods, the projection method and the regularization method, to solve the variational inequality problem. Two numerical experiments are reported to verify the theories and to investigate the physics of type-II superconductivity.