Banach空间中广义向量混合变分不等式的扰动Levitin-Polyak适定性

Levitin-Polyak Well-posedness by Perturbations of a Generalized Vector Mixed Variational Inequality in Banach Spaces

Tykhonov适定性与Levitin-Polyak适定性在研究各类最优化问题和变分不等式算法的收敛性中起着重要的作用.近年来,随着向量优化问题的出现和日渐成熟,对适定性的研究也开始在向量优化问题中进行.首先,提出了Banach空间中广义向量混合变分不等式扰动Levitin-Polyak适定性的概念,研究了广义向量混合变分不等式扰动Levitin-Polyak适定性的度量性质.其次,定义了广义向量混合变分不等式的gap函数,建立了广义向量混合变分不等式的扰动Levitin-Polyak适定性与其对应的gap函数相关的极小化问题的适定性的等价关系.到目前为止还没有关于广义向量混合变分不等式的扰动Levitin-Polyak适定性的结果,因此研究此类问题是非常有意义的.

Tykhonov and Levitin-Polyak well-posedness play a central role in the study of optimization problems.Recently,vector optimization problems have been intensively developed.Many researchers have tried to study well-possedness for vector optimization problems.First,the notion of Levitin-Polyak well-posedness by perturbations of a generalized vector mixed variational inequality is introduced in Banach spaces and some metric characterizations for the Levitin-Polyak well-posedness by perturbations of a generalized vector mixed variational inequality are presented.Second,by using the gap function of a generalized vector mixed variational inequality,the equivalent relationship between the Levitin-Polyak well-posedness by perturbations of a generalized vector mixed variational inequality and the related minimizing problem are established.So far,there are no results about Levitin-Polyak well-posedness by perturbations of a generalized vector mixed variational inequality,so it is very interesting to study this problem.