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. 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. (2025). 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.

(2025). 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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