ABSTRACT

The paper introduces two new numerical methods for solving a variational inequality problem whose constraint set is expressed as the solution set of a monotone and Lipschitz-type equilibrium problem in a Hilbert space. We present how to combine regularization terms in an extragradient method and prove that the iterative sequences generated by the resulting methods converge strongly to a solution of equilibrium problem which solves the associated variational inequality problem. Theorems of strong convergence are analysed which are based on the incorporated Tikhonov regularization method for equilibrium problems. The first method is designed in the case where the Lipschitz-type constants of bifunction are known. While the second method can be implemented more easily without the prior knownledge of Lipschitz-type constants. The reason is that the second method have used a new stepsize rule whose computation is simple and easy to check at each step. Several numerical experiments are performed and they have demonstrated the effectiveness and the fast convergence of the new methods over existing methods.

摘要

本文介绍了求解变分不等式问题的两种新的数值方法，其约束集表示为希尔伯特空间中单调和Lipschitz型平衡问题的解集。我们介绍了如何在外梯度方法中组合正则化项，并证明由所得方法生成的迭代序列强烈收敛到解决相关变分不等式问题的平衡问题的解决方案。分析了强收敛定理，这些定理基于结合的Tikhonov正则化方法解决平衡问题。第一种方法是在双函数的 Lipschitz 型常数已知的情况下设计的。而第二种方法可以更容易地实现，而无需先验已知的 Lipschitz 型常数。原因是第二种方法使用了一个新的步长规则，其计算简单，每一步都易于检查。进行了几个数值实验，它们证明了新方法相对于现有方法的有效性和快速收敛性。