Pareto front explained

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] 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] [4]

Definition

The Pareto frontier, P(Y), may be more formally described as follows. Consider a system with function

f:XRm

, where X is a compact set of feasible decisions in the metric space

Rn

, and Y is the feasible set of criterion vectors in

Rm

, such that

Y=\{y\inRm:y=f(x),x\inX\}

.

We assume that the preferred directions of criteria values are known. A point

y\prime\prime\inRm

is preferred to (strictly dominates) another point

y\prime\inRm

, written as

y\prime\prime\succy\prime

. The Pareto frontier is thus written as:

P(Y)=\{y\prime\inY:\{y\prime\prime\inY:y\prime\prime\succy\prime,y\primey\prime\prime\}=\empty\}.

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

i(x
z
i=f

i)

where
i,
x
1
i,
x
2

\ldots,

i)
x
n
is the vector of goods, both for all i. The feasibility constraint is
m
\sum
i=1
i
x
j

=bj

for

j=1,\ldots,n

. To find the Pareto optimal allocation, we maximize the Lagrangian:

Li((x

k)
k,j

,(λk)k,(\muj)

i(x
j)=f
m
k=2

λk(zk-fk(x

n
j=1

\muj\left(bj-\sum

m
k=1
k
x
j

\right)

where

(λk)k

and

(\muj)j

are the vectors of multipliers. Taking the partial derivative of the Lagrangian with respect to each good
k
x
j
for

j=1,\ldots,n

and

k=1,\ldots,m

gives the following system of first-order conditions:
\partialLi
\partial
i
x
j

=

1-\mu
f
j=0forj=1,\ldots,n,
\partialLi
\partial
k
x
j

=k

i-\mu
f
j=0

fork=2,\ldots,mandj=1,\ldots,n,

where

f
i
x
j
denotes the partial derivative of

f

with respect to
i
x
j
. Now, fix any

ki

and

j,s\in\{1,\ldots,n\}

. The above first-order condition imply that
i
f
i
x
j
=
i
f
i
x
s
\muj=
\mus
k
f
k
x
j
k
f
k
x
s

.

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

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:

\epsilon

-constraints method"[12] [13]

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

References

  1. Web site: proximedia. Pareto Front. 2018-10-08. www.cenaero.be. https://web.archive.org/web/20200226003108/https://www.cenaero.be/Page.asp?docid=27103&. 2020-02-26. dead.
  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. Book: Just, Richard E.. The welfare economics of public policy : a practical approach to project and policy evaluation. E. Elgar. Hueth, Darrell L., Schmitz, Andrew.. 2004. 1-84542-157-4. Cheltenham, UK. 18–21. 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. 10.3390/en6031439. free. 2117/18257. free.
  7. Nielsen. Frank. 1996. Output-sensitive peeling of convex and maximal layers. Information Processing Letters. 59. 5. 255–9. 10.1.1.259.1042. 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. 10.1145/321906.321910. 2698043. free.
  9. Godfrey. P.. Shipley. R.. Gryz. J.. 2006. Algorithms and Analyses for Maximal Vector Computation. VLDB Journal. 16. 5–28. 10.1.1.73.6344. 10.1007/s00778-006-0029-7. 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. 10.1007/s00158-005-0557-6. 1615-147X. 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. 10.1007/s00158-009-0460-7. 1615-147X. 122325484.
  12. 1971. 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. 10.1109/TSMC.1971.4308298. 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. 10.1016/j.amc.2009.03.037. 0096-3003.
  14. Legriel. Julien. Le Guernic. Colas. Cotton. Scott. Maler. Oded. 2010. Esparza. Javier. Majumdar. Rupak. Approximating the Pareto Front of Multi-criteria Optimization Problems. Tools and Algorithms for the Construction and Analysis of Systems. Lecture Notes in Computer Science. 6015 . en. Berlin, Heidelberg. Springer. 69–83. 10.1007/978-3-642-12002-2_6. 978-3-642-12002-2. free.