Title: On Lagrange multipliers of the KKT system in Hilbert spaces Show Chinese title

Title: 关于Hilbert空间中KKT系统的拉格朗日乘子

Abstract: In this paper we develop a new decomposition framework to deal with Lagrange multipliers of the Karush-Kuhn-Tucker (KKT) system of constrained optimization problems and variational inequalities in Hilbert spaces. It is different from existing frameworks based on separation theorems. We introduce the essential Lagrange multiplier and establish the basic theory of this new multiplier. The essential Lagrange multiplier poses essentially different existence results in finite and infinite-dimensional spaces. It can also be used to give an essential characterization of the convergence of multipliers generated by the classical augmented Lagrangian method. Our analysis reveals that the essential Lagrange multiplier is at the core of both theories and applications of Lagrange multipliers. Show Chinese abstract

Abstract: 在本文中，我们开发了一种新的分解框架，以处理约束优化问题和希尔伯特空间中变分不等式的KKT系统的拉格朗日乘子。它不同于基于分离定理的现有框架。我们引入了基本的拉格朗日乘子，并建立了这一新乘子的基本理论。基本的拉格朗日乘子在有限维和无限维空间中具有本质上不同的存在结果。它还可以用于给出经典增广拉格朗日方法生成的乘子收敛性的基本特征。我们的分析表明，基本的拉格朗日乘子是拉格朗日乘子理论和应用的核心。