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From Wikipedia, the free encyclopedia
Set of all Pareto efficient situations
In
multi-objective optimization
, the
Pareto front
(also called
Pareto frontier
or
Pareto curve
) is the set of all
Pareto efficient
solutions.
[
1
]
Colloquially, this means when there are many distinct objectives to consider in an
optimization problem
, a Pareto front represents the set of solutions where no solution outperforms any other solution in the set at every objective, and every solution not in the set is outperformed by at least one solution in the Pareto front in every objective.
[
2
]
The concept is widely used in
engineering
.
[
3
]
:
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.
[
4
]
:
63–65
[
5
]
:
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.
Definition
[
edit
]
The Pareto frontier,
P
(
Y
), may be more formally described as follows. Consider a system with function
f
:
X
→
R
m
{\displaystyle f:X\rightarrow \mathbb {R} ^{m}}
, where
X
is a
compact set
of feasible decisions in the
metric space
R
n
{\displaystyle \mathbb {R} ^{n}}
, and
Y
is the feasible set of criterion vectors in
R
m
{\displaystyle \mathbb {R} ^{m}}
, such that
Y
=
{
y
∈
R
m
:
y
=
f
(
x
)
,
x
∈
X
}
{\displaystyle Y=\{y\in \mathbb {R} ^{m}:\;y=f(x),x\in X\;\}}
.
We assume that the preferred directions of criteria values are known. A point
y
′
′
∈
R
m
{\displaystyle y^{\prime \prime }\in \mathbb {R} ^{m}}
is preferred to (strictly dominates) another point
y
′
∈
R
m
{\displaystyle y^{\prime }\in \mathbb {R} ^{m}}
, written as
y
′
′
≻
y
′
{\displaystyle y^{\prime \prime }\succ y^{\prime }}
. The Pareto frontier is thus written as:
P
(
Y
)
=
{
y
′
∈
Y
:
{
y
′
′
∈
Y
:
y
′
′
≻
y
′
,
y
′
≠
y
′
′
}
=
∅
}
.
{\displaystyle P(Y)=\{y^{\prime }\in Y:\;\{y^{\prime \prime }\in Y:\;y^{\prime \prime }\succ y^{\prime },y^{\prime }\neq y^{\prime \prime }\;\}=\emptyset \}.}
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.
[
6
]
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
)
{\displaystyle z_{i}=f^{i}(x^{i})}
where
x
i
=
(
x
1
i
,
x
2
i
,
…
,
x
n
i
)
{\displaystyle x^{i}=(x_{1}^{i},x_{2}^{i},\ldots ,x_{n}^{i})}
is the vector of goods, both for all
i
. The feasibility constraint is
∑
i
=
1
m
x
j
i
=
b
j
{\displaystyle \sum _{i=1}^{m}x_{j}^{i}=b_{j}}
for
j
=
1
,
…
,
n
{\displaystyle j=1,\ldots ,n}
. To find the Pareto optimal allocation, we maximize the
Lagrangian
:
L
i
(
(
x
j
k
)
k
,
j
,
(
λ
k
)
k
,
(
μ
j
)
j
)
=
f
i
(
x
i
)
+
∑
k
=
2
m
λ
k
(
z
k
−
f
k
(
x
k
)
)
+
∑
j
=
1
n
μ
j
(
b
j
−
∑
k
=
1
m
x
j
k
)
{\displaystyle L_{i}((x_{j}^{k})_{k,j},(\lambda _{k})_{k},(\mu _{j})_{j})=f^{i}(x^{i})+\sum _{k=2}^{m}\lambda _{k}(z_{k}-f^{k}(x^{k}))+\sum _{j=1}^{n}\mu _{j}\left(b_{j}-\sum _{k=1}^{m}x_{j}^{k}\right)}
where
(
λ
k
)
k
{\displaystyle (\lambda _{k})_{k}}
and
(
μ
j
)
j
{\displaystyle (\mu _{j})_{j}}
are the vectors of multipliers. Taking the
partial derivative
of the Lagrangian with respect to each good
x
j
k
{\displaystyle x_{j}^{k}}
for
j
=
1
,
…
,
n
{\displaystyle j=1,\ldots ,n}
and
k
=
1
,
…
,
m
{\displaystyle k=1,\ldots ,m}
gives the following system of first-order conditions:
∂
L
i
∂
x
j
i
=
f
x
j
i
1
−
μ
j
=
0
for
j
=
1
,
…
,
n
,
{\displaystyle {\frac {\partial L_{i}}{\partial x_{j}^{i}}}=f_{x_{j}^{i}}^{1}-\mu _{j}=0{\text{ for }}j=1,\ldots ,n,}
∂
L
i
∂
x
j
k
=
−
λ
k
f
x
j
k
i
−
μ
j
=
0
for
k
=
2
,
…
,
m
and
j
=
1
,
…
,
n
,
{\displaystyle {\frac {\partial L_{i}}{\partial x_{j}^{k}}}=-\lambda _{k}f_{x_{j}^{k}}^{i}-\mu _{j}=0{\text{ for }}k=2,\ldots ,m{\text{ and }}j=1,\ldots ,n,}
where
f
x
j
i
{\displaystyle f_{x_{j}^{i}}}
denotes the partial derivative of
f
{\displaystyle f}
with respect to
x
j
i
{\displaystyle x_{j}^{i}}
. Now, fix any
k
≠
i
{\displaystyle k\neq i}
and
j
,
s
∈
{
1
,
…
,
n
}
{\displaystyle j,s\in \{1,\ldots ,n\}}
. The above first-order condition imply that
f
x
j
i
i
f
x
s
i
i
=
μ
j
μ
s
=
f
x
j
k
k
f
x
s
k
k
.
{\displaystyle {\frac {f_{x_{j}^{i}}^{i}}{f_{x_{s}^{i}}^{i}}}={\frac {\mu _{j}}{\mu _{s}}}={\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.
[
7
]
Computation
[
edit
]
Algorithms
for computing the Pareto frontier of a
finite set
of alternatives have been studied in
computer science
and power engineering.
[
8
]
They include:
“The
maxima of a point set
”
“The maximum vector problem” or the
skyline query
[
9
]
[
10
]
[
11
]
“The scalarization algorithm” or the method of weighted sums
[
12
]
[
13
]
“The
ϵ
{\displaystyle \epsilon }
-constraints method”
[
14
]
[
15
]
[
16
]
Multi-objective Evolutionary Algorithms
[
17
]
[
18
]
Approximations
[
edit
]
Since generating the entire Pareto front is often computationally-hard, there are algorithms for computing an approximate Pareto-front. For example, Legriel et al.
[
19
]
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
[
20
]
compare several algorithms for Pareto-set approximations on various criteria, such as invariance to scaling, monotonicity, and computational complexity.
References
[
edit
]
↑
proximedia.
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.
www.cenaero.be
. Archived from
the original
on 2020-02-26
. Retrieved
2018-10-08
.
↑
Kang, Shida; Li, Kaiwen; Wang, Rui (2025-06-01).
“A survey on pareto front learning for multi-objective optimization”
.
Journal of Membrane Computing
.
7
(2):
128–
134.
doi
:
10.1007/s41965-024-00170-z
.
ISSN
2523-8914
.
↑
Goodarzi, E., Ziaei, M., & Hosseinipour, E. Z.,
Introduction to Optimization Analysis in Hydrosystem Engineering
(
Berlin
/
Heidelberg
:
Springer
, 2014),
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.
↑
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Multi-criteria Decision Analysis
, 2nd ed. (
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:
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, 2013),
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.
↑
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),
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.
↑
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18–
21.
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.
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↑
Just, Richard E.; Hueth, Darrell L.; Schmitz, Andrew (2005-01-01).
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. Edward Elgar Publishing.
ISBN
978-1-84542-157-1
.
↑
Tomoiagă, Bogdan; Chindriş, Mircea; Sumper, Andreas; Sudria-Andreu, Antoni; Villafafila-Robles, Roberto (2013).
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.
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.
6
(3):
1439–
55.
doi
:
10.3390/en6031439
.
hdl
:
2117/18257
.
↑
Nielsen, Frank (1996). “Output-sensitive peeling of convex and maximal layers”.
Information Processing Letters
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59
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CiteSeerX
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:
10.1016/0020-0190(96)00116-0
.
↑
Kung, H. T.; Luccio, F.; Preparata, F.P. (1975).
“On finding the maxima of a set of vectors”
.
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22
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S2CID
2698043
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Godfrey, P.; Shipley, R.; Gryz, J. (2006). “Algorithms and Analyses for Maximal Vector Computation”.
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Kim, I. Y.; de Weck, O. L. (2005). “Adaptive weighted sum method for multiobjective optimization: a new method for Pareto front generation”.
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.
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.
↑
Marler, R. Timothy; Arora, Jasbir S. (2009). “The weighted sum method for multi-objective optimization: new insights”.
Structural and Multidisciplinary Optimization
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41
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862.
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10.1007/s00158-009-0460-7
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ISSN
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.
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.
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↑
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↑
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↑
Legriel, Julien; Le Guernic, Colas; Cotton, Scott; Maler, Oded (2010). “Approximating the Pareto Front of Multi-criteria Optimization Problems”. In Esparza, Javier; Majumdar, Rupak (eds.).
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. Lecture Notes in Computer Science. Vol.
6015. Berlin, Heidelberg: Springer. pp.
69–
83.
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.
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.
↑
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“Quality Assessment of Pareto Set Approximations”
, in Branke, Jürgen; Deb, Kalyanmoy; Miettinen, Kaisa; Słowiński, Roman (eds.),
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373–
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, retrieved
2021-10-08
Retrieved from ”
https://en.wikipedia.org/w/index.php?title=Pareto_front&oldid=1365854598
”
Categories
:
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