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
Definition
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:
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.[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]
Computation
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.
^Jahan, A., Edwards, K. L., & Bahraminasab, M., Multi-criteria Decision Analysis, 2nd ed. (
Amsterdam:
Elsevier, 2013),
pp. 63–65.
^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.
^Kim, I. Y.; de Weck, O. L. (2005). "Adaptive weighted sum method for multiobjective optimization: a new method for Pareto front generation". Structural and Multidisciplinary Optimization. 31 (2): 105–116.
doi:
10.1007/s00158-005-0557-6.
ISSN1615-147X.
S2CID18237050.
^Marler, R. Timothy; Arora, Jasbir S. (2009). "The weighted sum method for multi-objective optimization: new insights". Structural and Multidisciplinary Optimization. 41 (6): 853–862.
doi:
10.1007/s00158-009-0460-7.
ISSN1615-147X.
S2CID122325484.
^"On a Bicriterion Formulation of the Problems of Integrated System Identification and System Optimization". IEEE Transactions on Systems, Man, and Cybernetics. SMC-1 (3): 296–297. 1971.
doi:
10.1109/TSMC.1971.4308298.
ISSN0018-9472.
^Mavrotas, George (2009). "Effective implementation of the ε-constraint method in Multi-Objective Mathematical Programming problems". Applied Mathematics and Computation. 213 (2): 455–465.
doi:
10.1016/j.amc.2009.03.037.
ISSN0096-3003.