Variational Inequalities Governed By Strongly Pseudomonotone Operators

TL;DR

This paper studies variational inequalities with strongly pseudomonotone operators, establishing error bounds, convergence of gradient projection methods, and analyzing convergence rates with numerical comparisons.

Contribution

It introduces a global error bound for solutions, proves convergence of gradient projection methods under boundedness, and compares algorithms through numerical experiments.

Findings

Global error bound established for solution set

Gradient projection method converges under boundedness

Numerical experiments compare algorithm performance

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.