Pareto front

In multi-objective optimization, the Pareto front (also called Pareto frontier or Pareto curve) is the set of all Pareto efficient solutions.

[1] The concept is widely used in engineering.

[2]: 111–148  It allows the designer to restrict attention to the set of efficient choices, and to make tradeoffs within this set, rather than considering the full range of every parameter.

[3]: 63–65 [4]: 399–412 The Pareto frontier, P(Y), may be more formally described as follows.

Consider a system with function

, where X is a compact set of feasible decisions in the metric space

, and Y is the feasible set of criterion vectors in

We assume that the preferred directions of criteria values are known.

A point

is preferred to (strictly dominates) another point

The Pareto frontier is thus written as: A significant aspect of the Pareto frontier in economics is that, at a Pareto-efficient allocation, the marginal rate of substitution is the same for all consumers.

[5] A formal statement can be derived by considering a system with m consumers and n goods, and a utility function of each consumer as

is the vector of goods, both for all i.

The feasibility constraint is

To find the Pareto optimal allocation, we maximize the Lagrangian: where

λ

μ

are the vectors of multipliers.

Taking the partial derivative of the Lagrangian with respect to each good

gives the following system of first-order conditions: where

denotes the partial derivative of

Now, fix any

The above first-order condition imply that Thus, in a Pareto-optimal allocation, the marginal rate of substitution must be the same for all consumers.

[citation needed] Algorithms for computing the Pareto frontier of a finite set of alternatives have been studied in computer science and power engineering.

[6] They include: Since generating the entire Pareto front is often computationally-hard, there are algorithms for computing an approximate Pareto-front.

For example, Legriel et al.[17] call a set S an ε-approximation of the Pareto-front P, if the directed Hausdorff distance between S and P is at most ε.

They observe that an ε-approximation of any Pareto front P in d dimensions can be found using (1/ε)d queries.

Zitzler, Knowles and Thiele[18] compare several algorithms for Pareto-set approximations on various criteria, such as invariance to scaling, monotonicity, and computational complexity.

Example of a Pareto frontier. The boxed points represent feasible choices, and smaller values are preferred to larger ones. Point C is not on the Pareto frontier because it is dominated by both point A and point B . Points A and B are not strictly dominated by any other, and hence lie on the frontier.
A production-possibility frontier . The red line is an example of a Pareto-efficient frontier, where the frontier and the area left and below it are a continuous set of choices. The red points on the frontier are examples of Pareto-optimal choices of production. Points off the frontier, such as N and K, are not Pareto-efficient, since there exist points on the frontier which Pareto-dominate them.