AI 中文总结
针对现有方法（如NNC、NBI）无法捕获完整Pareto前沿的问题，提出广义法向约束（GNC）方法，通过几何框架实现100%捕获，适用于n目标优化问题。
AI 中文摘要
本文提出了一个用于生成完整Pareto前沿的综合几何与计算框架。现有几种方法在结构上无法捕获完整的可允许Pareto区域，包括广泛使用的加权和法、折衷规划法、法向边界交点（NBI）法和归一化法向约束（NNC）法。NNC和NBI共享相同的Pareto生成网格构造，对于三目标问题，它们在结构上无法捕获50%的可允许Pareto区域。更一般地，对于n目标问题，可允许捕获分数按阶乘递减为1/(n-1)!，相应的遗漏分数增加到1-1/(n-1)!。相比之下，本文新开发的广义法向约束（GNC）方法在结构上能够捕获100%的可允许Pareto区域。所提出的GNC方法针对一般的n目标优化问题进行了公式化，并通过统一的几何、数学和计算框架进行开发，辅以有洞察力的示例。多目标优化在广泛的应用中扮演重要角色，包括经济学、产品设计和工程管理。因此，优化方法生成覆盖完整Pareto前沿的代表性子集的能力至关重要。
英文摘要
This paper presents a unified geometric, mathematical, and computational framework for the generation of the $complete$ admissible Pareto frontier. Several existing methods are structurally unable to capture the complete admissible Pareto frontier. These include widely used methods such as the weighted sum, the Normal Boundary Intersection (NBI) method, and the Normalized Normal Constraint (NNC) method. NNC and NBI, which share the same Pareto-generation grid construction, are structurally unable to capture 50% of the admissible Pareto region for tri-objective problems. More generally, for an $n$-objective problem, the admissible capture fraction decreases factorially as $1/(n-1)!$, and the corresponding missed fraction increases to $1-1/(n-1)!$. By contrast, the newly developed Generalized Normal Constraint (GNC) method introduced in the present work is structurally capable of capturing the complete admissible Pareto frontier. The proposed GNC method is formulated for general $n$-objective optimization problems and is developed through a unified geometric, mathematical, and computational framework supported by computational examples. Multiobjective optimization plays an important role in a broad range of applications, including economics, product design, and engineering management. Accordingly, the ability of a Pareto-generation method to generate a representative subset spanning the $complete$ admissible Pareto frontier is of fundamental importance for multiobjective optimization.
Comments49 pages, 13 figures
