Two modifications of the inertial Tseng extragradient method with self-adaptive step size for solving monotone variational inequality problems

Lateef Olakunle Jolaoso,Timilehin Opeyemi Alakoya,Oluwatosin Temitope Mewomo

doi:10.1515/dema-2020-0013

Lateef Olakunle Jolaoso, Timilehin Opeyemi Alakoya + Show 1 more

Open Access

https://doi.org/10.1515/dema-2020-0013

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Journal: Demonstratio Mathematica	Publication Date: Sep 15, 2020
Citations: 25	License type: CC BY 4.0

Affiliation: University of KwaZulu-Natal

Abstract

AbstractIn this work, we introduce two new inertial-type algorithms for solving variational inequality problems (VIPs) with monotone and Lipschitz continuous mappings in real Hilbert spaces. The first algorithm requires the computation of only one projection onto the feasible set per iteration while the second algorithm needs the computation of only one projection onto a half-space, and prior knowledge of the Lipschitz constant of the monotone mapping is not required in proving the strong convergence theorems for the two algorithms. Under some mild assumptions, we prove strong convergence results for the proposed algorithms to a solution of a VIP. Finally, we provide some numerical experiments to illustrate the efficiency and advantages of the proposed algorithms.

Highlights

Let H be a real Hilbert space with the inner product 〈⋅,⋅〉 and the induced norm ∥⋅∥
The solution set of variational inequality problems (VIPs) (1) is denoted by VI(C, A)
Motivated by the works of Yang et al [34] and Thong et al [35] and the current research interest in this direction, we propose two new inertial-type algorithms for solving the VIP (1) based on the Tseng extragradient method (TEGM) and Moudafi’s viscosity scheme which does not require a prior knowledge of the Lipschitz constant of the monotone operator

Summary

Introduction

Let H be a real Hilbert space with the inner product 〈⋅,⋅〉 and the induced norm ∥⋅∥. Let C be a nonempty, closed, and convex subset in H. Set n ≔ n + 1 and return to Step 1, where F : H → H is monotone and Lipschitz continuous with constant L > 0, f : H → H is a strict contraction mapping with constant ρ ∈ [0, 1), and {αn} ⊂ (0, 1) They proved the strong convergence of the algorithm without any prior knowledge of the Lipschitz constant of the mapping. Motivated by the works of Yang et al [34] and Thong et al [35] and the current research interest in this direction, we propose two new inertial-type algorithms for solving the VIP (1) based on the TEGM and Moudafi’s viscosity scheme which does not require a prior knowledge of the Lipschitz constant of the monotone operator. The numerical illustrations show that our proposed algorithms with inertial effects converge faster than the original algorithms without inertial effects

Preliminaries

Main results

Numerical experiments

Conclusion