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.

摘要

本文研究了希尔伯特空间上由强伪单调算子控制的变分不等式的定性和定量方面。首先，我们为给定问题的解集建立一个全局误差界，残差函数是法线贴图。其次，我们将证明当算子有界在约束集上时，梯度投影法（GPM）生成的迭代序列具有步长形成不可和的正实数递减序列，收敛于问题的唯一解。给出了两个反例来说明有界假设的必要性和步长的变化。我们还分析了该方法生成的迭代序列的收敛速度。最后，