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., where X is a compact set of feasible decisions in the metric space, and Y is the feasible set of criterion vectors inWe assume that the preferred directions of criteria values are known.is preferred to (strictly dominates) another pointThe 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 asThe feasibility constraint isTo find the Pareto optimal allocation, we maximize the Lagrangian: whereTaking the partial derivative of the Lagrangian with respect to each goodgives the following system of first-order conditions: wheredenotes the partial derivative ofThe 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.
