In multi-objective optimization, the Pareto front (also called Pareto frontier or Pareto curve) is the set of all Pareto efficient solutions. Informally, this means when there are many distinct objectives to consider in an optimization problem, a Pareto front represents the set of solutions where no solution outperforms any other solution due to trade-offs among the objectives, and that set excludes the remaining solutions which are outperformed. Outperforming is called Pareto dominance: a solution A dominates (outperforms) B if A is no worse than B in every objective, and better than B in at least one objective. The concept is widely used in engineering. 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.
Definition The Pareto frontier, P(Y), may be more formally described as follows. Consider a system with function f : X → R m {\displaystyle f:X\rightarrow \mathbb {R} ^{m}} , where X is a compact set of feasible decisions in the metric space R n {\displaystyle \mathbb {R} ^{n}} , and Y is the feasible set of criterion vectors in R m {\displaystyle \mathbb {R} ^{m}} , such that Y = { y ∈ R m : y = f ( x ) , x ∈ X } {\displaystyle Y=\{y\in \mathbb {R} ^{m}:\;y=f(x),x\in X\;\}} . We assume that the preferred directions of criteria values are known. A point y ′ ′ ∈ R m {\displaystyle y^{\prime \prime }\in \mathbb {R} ^{m}} is preferred to (strictly dominates) another point y ′ ∈ R m {\displaystyle y^{\prime }\in \mathbb {R} ^{m}} , written as y ′ ′ ≻ y ′ {\displaystyle y^{\prime \prime }\succ y^{\prime }} . The Pareto frontier is thus written as:
P ( Y ) = { y ′ ∈ Y : { y ′ ′ ∈ Y : y ′ ′ ≻ y ′ , y ′ ≠ y ′ ′ } = ∅ } . {\displaystyle P(Y)=\{y^{\prime }\in Y:\;\{y^{\prime \prime }\in Y:\;y^{\prime \prime }\succ y^{\prime },y^{\prime }\neq y^{\prime \prime }\;\}=\emptyset \}.}
Marginal rate of substitution 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. A formal statement can be derived by considering a system with m consumers and n goods, and a utility function of each consumer as z i = f i ( x i ) {\displaystyle z_{i}=f^{i}(x^{i})} where x i = ( x 1 i , x 2 i , … , x n i ) {\displaystyle x^{i}=(x_{1}^{i},x_{2}^{i},\ldots ,x_{n}^{i})} is the vector of goods, both for all i. The feasibility constraint is ∑ i = 1 m x j i = b j {\displaystyle \sum _{i=1}^{m}x_{j}^{i}=b_{j}} for j = 1 , … , n {\displaystyle j=1,\ldots ,n} . To find the Pareto optimal allocation, we maximize the Lagrangian:
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