论文标题
精确的增强拉格朗日人在希尔伯特空间中有限的优化问题I：理论
Exact augmented Lagrangians for constrained optimization problems in Hilbert spaces I: Theory
论文作者
论文摘要
在这项两部分的研究中，我们开发了一种普通理论，即所谓的精确增强拉格朗日人，以解决希尔伯特空间中的优化问题。与传统的非平滑惩罚功能相反，这些增强的拉格朗日人在平稳问题上不断差异，并且不受马拉托斯效应的影响，这使得它们特别吸引了数值优化的应用。我们的目的是介绍精确增强拉格朗日人的各种理论特性的详细研究，并讨论这些功能的几种应用，以限制变化问题，PDE限制问题以及最佳控制问题。 第一篇论文致力于对希尔伯特空间中优化问题的精确增强拉格朗日的理论分析。我们获得了这种增强的拉格朗日及其梯度的几个有用的估计，并为与本地/全球最佳的解决方案相对应的约束问题的KKT点呈现了几种类型的条件，该解决方案是确切的增强Lagrangian的本地/全局最小化。
In this two-part study, we develop a general theory of the so-called exact augmented Lagrangians for constrained optimization problems in Hilbert spaces. In contrast to traditional nonsmooth exact penalty functions, these augmented Lagrangians are continuously differentiable for smooth problems and do not suffer from the Maratos effect, which makes them especially appealing for applications in numerical optimization. Our aim is to present a detailed study of various theoretical properties of exact augmented Lagrangians and discuss several applications of these functions to constrained variational problems, problems with PDE constraints, and optimal control problems. The first paper is devoted to a theoretical analysis of an exact augmented Lagrangian for optimization problems in Hilbert spaces. We obtain several useful estimates of this augmented Lagrangian and its gradient, and present several types of sufficient conditions for KKT-points of a constrained problem corresponding to locally/globally optimal solutions to be local/global minimisers of the exact augmented Lagrangian.
