Editing Pareto efficiency (section)

== Pareto order ==
If multiple sub-goals <math>f_i</math> (with <math>i > 1</math>) exist, combined into a vector-valued objective function <math>\vec{f}=(f_1, \dots f_n)^T</math>, generally, finding a unique optimum <math>\vec{x}^* </math> becomes challenging. This is due to the absence of a [[total order]] relation for <math>n >1</math> which would not always prioritize one target over another target (like the [[lexicographical order]]). In the multi-objective optimization setting, various solutions can be "incomparable"<ref>"The main difficulty is that, in contrast to the single-objective case where there is a total order relation between solutions, Pareto dominance is a partial order, which leads to solutions (and solution sets) being incomparable" Li, M., López-Ibáñez, M., & Yao, X. (Accepted/In press). Multi-Objective Archiving. IEEE Transactions on Evolutionary Computation. https://arxiv.org/pdf/2303.09685.pdf</ref> as there is no total order relation to facilitate the comparison <math>\vec{f}(\vec{x}^*) \geq \vec{f}(\vec{x})</math>. Only the Pareto order is applicable:

Consider a vector-valued minimization problem: <math>\vec{y}^{(1)} \in \mathbb{R}^m</math> Pareto dominates <math>\vec{y}^{(2)} \in \mathbb{R}^m</math> if and only if:<ref name = "Emmerich2018">Emmerich, M.T.M., Deutz, A.H. A tutorial on multiobjective optimization: fundamentals and evolutionary methods. Nat Comput 17, 585–609 (2018). https://doi.org/10.1007/s11047-018-9685-y</ref> :<math>\forall i \in \{1,\dots m\}:\vec{y}_i^{(1)} \leq \vec{y}_i^{(2)}</math> and <math>\exists j \in \{1,\dots m\}:\vec{y}_j^{(1)} < \vec{y}_j^{(2)}.</math> We then write <math>\vec{y}^{(1)} \prec \vec{y}^{(2)}</math>, where <math>\prec</math> is the Pareto order. This means that <math>\vec{y}^{(1)}</math> is not worse than <math>\vec{y}^{(2)}</math> in any goal but is better (since smaller) in at least one goal <math>j</math>. The Pareto order is a strict [[partial order]], though it is not a [[product order]] (neither non-strict nor strict).

If<ref name="Emmerich2018" /> <math>\vec{f}(\vec{x}_1) \prec \vec{f}(\vec{x}_2)</math>, then this defines a [[preorder]] in the search space and we say <math>\vec{x}_1</math> Pareto dominates the alternative <math>\vec{x}_2</math> and we write <math>\vec{x}_1 \prec_{\vec{f}} \vec{x}_2</math>.

[[Image:Pareto order dominated.png|thumb|<math>\vec{f}(x)</math> dominates <math>\vec{f}(y)</math> in the Pareto order (which seeks to minimize the goals <math>f_1</math> and <math>f_2</math>).]] 
[[Image:Pareto order not dominated.png|thumb|<math>\vec{f}(x)</math> does not dominate <math>\vec{f}(y)</math> in the Pareto order and <math>\vec{f}(y)</math> does not dominate <math>\vec{f}(x)</math> in the Pareto order (which seeks to minimize the goals <math>f_1</math> and <math>f_2</math>).]]