Two Nonmonotonic Self-Adaptive Strongly Convergent Projection-Type Methods for Solving Pseudomonotone Variational Inequalities

Fanglei Wang,Chainarong Khunpanuk,Bancha Panyanak,Nuttapol Pakkaranang

doi:10.1155/2021/8327694

Fanglei Wang, Chainarong Khunpanuk + Show 2 more

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https://doi.org/10.1155/2021/8327694

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Abstract

The primary objective of this study is to introduce two novel extragradient-type iterative schemes for solving variational inequality problems in a real Hilbert space. The proposed iterative schemes extend the well-known subgradient extragradient method and are used to solve variational inequalities involving the pseudomonotone operator in real Hilbert spaces. The proposed iterative methods have the primary advantage of using a simple mathematical formula for step size rule based on operator information rather than the Lipschitz constant or another line search method. Strong convergence results for the suggested iterative algorithms are well-established for mild conditions, such as Lipschitz continuity and mapping monotonicity. Finally, we present many numerical experiments that show the effectiveness and superiority of iterative methods.

Highlights

The primary objective of this research is to investigate the iterative methodologies used to estimate the solution of variational inequalities in a real Hilbert space
A mapping L : Z ⟶ Z is said to be pseudomonotone if hLðp1Þ, p2 − p1i ≥ 0 ⇒ hLðp2Þ, p1 − p2i ≤ 0, ∀p1, p2 ∈ AðPMÞ: ð1Þ
In this part of the research article, we propose two new methods and the corresponding strong convergence theorems

Summary

Introduction

The primary objective of this research is to investigate the iterative methodologies used to estimate the solution of variational inequalities in a real Hilbert space. To establish the convergence analysis theorems, the following conditions need to be satisfied: Condition 1. The solution set of the problem (VIP) denoted by Ω and it is nonempty. A mapping L : Z ⟶ Z is said to be pseudomonotone if hLðp1Þ, p2 − p1i ≥ 0 ⇒ hLðp2Þ, p1 − p2i ≤ 0, ∀p1, p2 ∈ AðPMÞ: ð1Þ. A mapping L : Z ⟶ Z is said to be Lipschitz continuous with constant L > 0 if kLðp1Þ − Lðp2Þk ≤ Lkp1 − p2k, ∀p1, p2 ∈ A ðLCÞ: ð2Þ

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