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.