Pareto front

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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:

Marginal rate of substitution[edit]

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 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.


  1. ^ proximedia. "Pareto Front". Retrieved 2018-10-08.
  2. ^ Goodarzi, E., Ziaei, M., & Hosseinipour, E. Z., Introduction to Optimization Analysis in Hydrosystem Engineering (Berlin/Heidelberg: Springer, 2014), pp. 111–148.
  3. ^ Jahan, A., Edwards, K. L., & Bahraminasab, M., Multi-criteria Decision Analysis, 2nd ed. (Amsterdam: Elsevier, 2013), pp. 63–65.
  4. ^ 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.
  5. ^ Just, Richard E. (2004). The welfare economics of public policy : a practical approach to project and policy evaluation. Hueth, Darrell L., Schmitz, Andrew. Cheltenham, UK: E. Elgar. pp. 18–21. ISBN 1-84542-157-4. OCLC 58538348.
  6. ^ Tomoiagă, Bogdan; Chindriş, Mircea; Sumper, Andreas; Sudria-Andreu, Antoni; Villafafila-Robles, Roberto (2013). "Pareto Optimal Reconfiguration of Power Distribution Systems Using a Genetic Algorithm Based on NSGA-II". Energies. 6 (3): 1439–55. doi:10.3390/en6031439. hdl:2117/18257.
  7. ^ Nielsen, Frank (1996). "Output-sensitive peeling of convex and maximal layers". Information Processing Letters. 59 (5): 255–9. CiteSeerX doi:10.1016/0020-0190(96)00116-0.
  8. ^ Kung, H. T.; Luccio, F.; Preparata, F.P. (1975). "On finding the maxima of a set of vectors". Journal of the ACM. 22 (4): 469–76. doi:10.1145/321906.321910. S2CID 2698043.
  9. ^ Godfrey, P.; Shipley, R.; Gryz, J. (2006). "Algorithms and Analyses for Maximal Vector Computation". VLDB Journal. 16: 5–28. CiteSeerX doi:10.1007/s00778-006-0029-7. S2CID 7374749.
  10. ^ 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. ISSN 1615-147X. S2CID 18237050.
  11. ^ 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. ISSN 1615-147X. S2CID 122325484.
  12. ^ "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. ISSN 0018-9472.
  13. ^ 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. ISSN 0096-3003.
  14. ^ Legriel, Julien; Le Guernic, Colas; Cotton, Scott; Maler, Oded (2010). Esparza, Javier; Majumdar, Rupak (eds.). "Approximating the Pareto Front of Multi-criteria Optimization Problems". Tools and Algorithms for the Construction and Analysis of Systems. Lecture Notes in Computer Science. Berlin, Heidelberg: Springer. 6015: 69–83. doi:10.1007/978-3-642-12002-2_6. ISBN 978-3-642-12002-2.
  15. ^ Zitzler, Eckart; Knowles, Joshua; Thiele, Lothar (2008), Branke, Jürgen; Deb, Kalyanmoy; Miettinen, Kaisa; Słowiński, Roman (eds.), "Quality Assessment of Pareto Set Approximations", Multiobjective Optimization: Interactive and Evolutionary Approaches, Lecture Notes in Computer Science, Berlin, Heidelberg: Springer, pp. 373–404, doi:10.1007/978-3-540-88908-3_14, ISBN 978-3-540-88908-3, retrieved 2021-10-08