Pareto Frontier

A Pareto Frontier is a Compact Space that ...

• AKA: Pareto Efficiency.
• See: Metric Space, Compact Space.

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References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Pareto_efficiency#Pareto_frontier Retrieved:2015-7-18.
  • For a given system, the Pareto frontier or Pareto set is the set of parameterizations (allocations) that are all Pareto efficient. Finding Pareto frontiers is particularly useful in engineering. By yielding all of the potentially optimal solutions, a designer can make focused tradeoffs within this constrained set of parameters, rather than needing to consider the full ranges of parameters.

    The Pareto frontier, P(Y), may be more formally described as follows. Consider a system with function [math]\displaystyle{ f: \mathbb{R}^n \rightarrow \mathbb{R}^m }[/math] , where X is a compact set of feasible decisions in the metric space [math]\displaystyle{ \mathbb{R}^n }[/math] , and Y is the feasible set of criterion vectors in [math]\displaystyle{ \mathbb{R}^m }[/math] , such that [math]\displaystyle{ Y = \{ y \in \mathbb{R}^m:\; y = f(x), x \in X\;\} }[/math] .

    We assume that the preferred directions of criteria values are known. A point [math]\displaystyle{ y^{\prime\prime} \in \mathbb{R}^m\; }[/math] is preferred to (strictly dominating) another point [math]\displaystyle{ y^{\prime} \in \mathbb{R}^m\; }[/math] , written as [math]\displaystyle{ y^{\prime\prime} \succ y^{\prime} }[/math] . The Pareto frontier is thus written as: [math]\displaystyle{ P(Y) = \{ y^{\prime} \in Y: \; \{y^{\prime\prime} \in Y:\; y^{\prime\prime} \succ y^{\prime}, y^{\prime\prime} \neq y^{\prime} \; \} = \empty \} }[/math] .

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