利用诺特定理构造帕累托集
Constructing Pareto Sets Using Noether's Second Theorem
- MIREA – Russian Technological University（莫斯科信息俄罗斯国立技术大学）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文基于诺特第二定理，建立系统规范对称性与帕累托集结构的联系，提出规范正则化方法以降低解空间维数，并通过线性二次调节器和ZDT1问题验证。
中文摘要 AI 辅助
本文提出了一种基于诺特第二定理的分析方法，用于构造多目标变分控制问题中的帕累托集。文中建立了描述被控对象的动力系统的规范对称性与帕累托集结构之间的基本联系。研究表明，帕累托前沿可以解释为系统及质量准则在对称群作用下不变性所导出的守恒律。本文提出了一种规范正则化概念，该概念允许在不改变帕累托集的情况下消除不影响准则的变量，从而降低解空间的维数。文中给出了具有两个冲突准则的线性二次调节器以及标准ZDT1测试问题的数值算例。
英文摘要
This paper presents an analytical approach to constructing the Pareto set in multiobjective variational control problems, based on Noether's second theorem. A fundamental connection is established between the gauge symmetries of the dynamical system describing the plant and the structure of the Pareto set. It is shown that the Pareto front can be interpreted as a conservation law arising from the invariance of the system and the quality criteria with respect to a symmetry group. A gauge regularization concept is proposed, which allows eliminating variables that do not affect the criteria without changing the Pareto set, thereby reducing the dimension of the solution space. Numerical examples are provided for the linear-quadratic regulator with two conflicting criteria and for the standard ZDT1 test problem.
