ParetoFrontier

The Pareto frontier, also known as the Pareto front, is a concept in multi-objective optimization and welfare economics. It comprises the set of Pareto-optimal solutions, where a solution is Pareto-optimal if no other feasible solution is at least as good in all objectives and strictly better in at least one. In other words, the frontier contains all non-dominated outcomes when evaluating multiple criteria.

Points on the frontier are efficient trade-offs. A point not on the frontier is dominated by at least one frontier point, meaning there exists another solution that is better in all objectives or strictly better in at least one while not worse in others. The frontier can be curved or even disconnected depending on the problem; in convex problems it is often connected, and in linear problems it is piecewise linear.

Computing the frontier involves methods such as scalarization (turning multiple objectives into a single objective via weights), the epsilon-constraint method, and multi-objective evolutionary algorithms (for example NSGA-II, SPEA2). In practice, the frontier is often approximated by sampling feasible solutions and retaining non-dominated ones.

Applications span engineering design, economics, finance, logistics, and environmental planning. The frontier provides a set of acceptable compromises, allowing decision makers to select a preferred point based on preferences, risk, or external constraints.

Historically, the term Pareto efficiency and the frontier are named after Vilfredo Pareto, who studied the distribution of resources and welfare in social systems in the 19th century.