Set of all Pareto efficient situations From Wikipedia, the free encyclopedia
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
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.
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 , such that .
We assume that the preferred directions of criteria values are known. A point is preferred to (strictly dominates) another point , written as . 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 where is the vector of goods, both for all i. The feasibility constraint is for . To find the Pareto optimal allocation, we maximize the Lagrangian:
where and are the vectors of multipliers. Taking the partial derivative of the Lagrangian with respect to each good for and gives the following system of first-order conditions:
where denotes the partial derivative of with respect to . Now, fix any and . 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.[14] 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[15] compare several algorithms for Pareto-set approximations on various criteria, such as invariance to scaling, monotonicity, and computational complexity.
Goodarzi, E., Ziaei, M., & Hosseinipour, E. Z., Introduction to Optimization Analysis in Hydrosystem Engineering (Berlin/Heidelberg: Springer, 2014), pp. 111–148.
Costa, N. R., & Lourenço, J. A., "Exploring Pareto Frontiers in the Response Surface Methodology", in G.-C. Yang, S.-I. Ao, & L. Gelman, eds., Transactions on Engineering Technologies: World Congress on Engineering 2014 (Berlin/Heidelberg: Springer, 2015), pp. 399–412.
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