Pareto front

 ( Power Engineering ) 

In multi-objective optimization, the Pareto front (also called Pareto frontier or Pareto curve) is the set of all Pareto efficient solutions. Colloquially, this means when there are many distinct factors to consider in an optimization problem, a Pareto front represents the set of solutions that are "equally good" overall, albeit by making different concessions and compromises. The concept is widely used in engineering.Goodarzi, E., Ziaei, M., & Hosseinipour, E. Z., Introduction to Optimization Analysis in Hydrosystem Engineering (Berlin/Heidelberg: Springer, 2014), pp. 111–148. It allows the designer to restrict attention to the set of efficient choices, and to make Trade-off within this set, rather than considering the full range of every parameter.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.

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Definition The Pareto frontier, P( Y), may be more formally described as follows. Consider a system with function $f:\; X\; \backslash rightarrow\; \backslash mathbb\{R\}^m$, where X is a Compact space of feasible decisions in the metric space $\backslash mathbb\{R\}^n$, and Y is the feasible set of criterion vectors in $\backslash mathbb\{R\}^m$, such that $Y\; =\; \backslash \{\; y\; \backslash in\; \backslash mathbb\{R\}^m:\backslash ;\; y\; =\; f(x),\; x\; \backslash in\; X\backslash ;\backslash \}$.

We assume that the preferred directions of criteria values are known. A point $y^\{\backslash prime\backslash prime\}\; \backslash in\; \backslash mathbb\{R\}^m$ is preferred to (strictly dominates) another point $y^\{\backslash prime\}\; \backslash in\; \backslash mathbb\{R\}^m$, written as $y^\{\backslash prime\backslash prime\}\; \backslash succ\; y^\{\backslash prime\}$. The Pareto frontier is thus written as:

$P(Y)\; =\; \backslash \{\; y^\backslash prime\; \backslash in\; Y:\; \backslash ;\; \backslash \{y^\{\backslash prime\backslash prime\}\; \backslash in\; Y:\backslash ;\; y^\{\backslash prime\backslash prime\}\; \backslash succ\; y^\{\backslash prime\},\; y^\backslash prime\; \backslash neq\; y^\{\backslash prime\backslash prime\}\; \backslash ;\; \backslash \}\; =\; \backslash empty\; \backslash \}.$

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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.

Just, Richard E. (2026). 9781845421571, E. Elgar. ISBN 9781845421571

A formal statement can be derived by considering a system with m consumers and n goods, and a utility function of each consumer as $z\_i=f^i(x^i)$ where $x^i=(x\_1^i,\; x\_2^i,\; \backslash ldots,\; x\_n^i)$ is the vector of goods, both for all i. The feasibility constraint is $\backslash sum\_\{i=1\}^m\; x\_j^i\; =\; b\_j$ for $j=1,\backslash ldots,n$. To find the Pareto optimal allocation, we maximize the Lagrangian:

$L\_i((x\_j^k)\_\{k,j\},\; (\backslash lambda\_k)\_k,\; (\backslash mu\_j)\_j)=f^i(x^i)+\backslash sum\_\{k=2\}^m\; \backslash lambda\_k(z\_k-\; f^k(x^k))+\backslash sum\_\{j=1\}^n\; \backslash mu\_j\; \backslash left(\; b\_j-\backslash sum\_\{k=1\}^m\; x\_j^k\; \backslash right)$

where $(\backslash lambda\_k)\_k$ and $(\backslash mu\_j)\_j$ are the vectors of multipliers. Taking the partial derivative of the Lagrangian with respect to each good $x\_j^k$ for $j=1,\backslash ldots,n$ and $k=1,\backslash ldots,\; m$ gives the following system of first-order conditions:

$\backslash frac\{\backslash partial\; L\_i\}\{\backslash partial\; x\_j^i\}\; =\; f\_\{x^i\_j\}^1-\backslash mu\_j=0\backslash text\{\; for\; \}j=1,\backslash ldots,n,$

$\backslash frac\{\backslash partial\; L\_i\}\{\backslash partial\; x\_j^k\}\; =\; -\backslash lambda\_k\; f\_\{x^k\_j\}^i-\backslash mu\_j=0\; \backslash text\{\; for\; \}k=\; 2,\backslash ldots,m\; \backslash text\{\; and\; \}j=1,\backslash ldots,n,$

where $f\_\{x^i\_j\}$ denotes the partial derivative of $f$ with respect to $x\_j^i$. Now, fix any $k\backslash neq\; i$ and $j,s\backslash in\; \backslash \{1,\backslash ldots,n\backslash \}$. The above first-order condition imply that

$\backslash frac\{f\_\{x\_j^i\}^i\}\{f\_\{x\_s^i\}^i\}=\backslash frac\{\backslash mu\_j\}\{\backslash mu\_s\}=\backslash frac\{f\_\{x\_j^k\}^k\}\{f\_\{x\_s^k\}^k\}.$

Thus, in a Pareto-optimal allocation, the marginal rate of substitution must be the same for all consumers.

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Computation Algorithm for computing the Pareto frontier of a finite set of alternatives have been studied in computer science and power engineering. They include:

• "The maxima of a point set"
• "The maximum vector problem" or the Skyline operator
• "The scalarization algorithm" or the method of weighted sums
• "The $\backslash epsilon$-constraints method"
• Multi-objective Evolutionary Algorithms

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Approximations Since generating the entire Pareto front is often computationally-hard, there are algorithms for computing an approximate Pareto-front. For example, Legriel et al.

(2026). 9783642120022, Springer. ISBN 9783642120022

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 compare several algorithms for Pareto-set approximations on various criteria, such as invariance to scaling, monotonicity, and computational complexity.

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