Variational inequalities governed by strongly pseudomonotone operators

ABSTRACT

Qualitative and quantitative aspects for variational inequalities governed by strongly pseudomonotone operators on Hilbert space are investigated in this paper. First, we establish a global error bound for the solution set of the given problem with the residual function being the normal map. Second, we will prove that the iterative sequences generated by gradient projection method (GPM) with stepsizes forming a non-summable diminishing sequence of positive real numbers converge to the unique solution of the problem when the operator is bounded over the constraint set. Two counter-examples are given to show the necessity of the boundedness assumption and the variation of stepsizes. We also analyze the convergence rate of the iterative sequences generated by this method. Finally, we give an in-depth comparison between our algorithm and a recent related algorithm through several numerical experiments.

Keywords:

• Variational inequalities
• strong pseudomonotonicity
• gradient projection method
• convergence
• convergence rate

Mathematics Subject Classification (2010):

• 47J20
• 49J40
• 49M30

Acknowledgements

We would like to thank Mr. Huynh Phuoc Truong for his comments and discussion in Example 4.1. We are also grateful to the anonymous referee and the associate editor for constructive comments and suggestions, which greatly improved the paper. Pham Duy Khanh was supported, in part, by the Fondecyt Postdoc Project 3180080, the Basal Program CMM–AFB 170001 from CONICYT–Chile, and the National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2017.325.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.