math - 关于优化中帕累托优势的正确表示法

Question

我搜索了 Pareto Dominance 的正确符号，但我不知道这是否是 Mathexchange 的问题或者在这里..

在多篇论文中，喜欢经典Deb nsga2 paper ，你可以找到这样写的帕累托优势关系

如果A支配B

在wikipedia ，或其他纸张，如 this one of Zitler ，你会发现另一个符号:

如果A支配B

这里最好和正确的数学符号是什么？这个符号的确切名称是什么？

Answer 1

≺ 称为 precedes , ≻ succeeds .

根据众所周知的Evolutionary Optimization Algorithms A 支配 B 的标准符号是 A ≻ B:

20.1 Pareto Optimality

[...]

Domination: a point x^* is said to dominate x if the following two conditions hold:

1. f_i(x^*) <= f_i(x) for all i ∈ [1,k]
2. f_i(x^*) < f_i(x) for at least one j ∈ [1,k]

that is x^* is at least as good as x for all objective function values and it's better than x for at least one objective function value. We use the notation:

x^* ≻ x

to indicate that x^* dominates x.

This notation can be confusing because the symbol ≻ looks like a "greater than" symbol but since we deal mainly with minimization problems, the symbol ≻ means the function values of x^* are less than or equal to those of x.

However this notation is standard in the literature, so this is the notation that we use.

然而，甚至使用了相反的表示法(可能是为了“避免”作者所指的混淆!)