Sample Pareto fronts

Kursawe

Proposed by Kursawe [4].

$\begin{displaymath}\textnormal{Minimize } f_{1}(\vec{x}) = \sum^{n-1}_{i=1}\le......exp \left( -0.2 \sqrt{x_{i}^{2} + x_{i+1}^{2}}\right) \right)\end{displaymath}$

(1)

$\begin{displaymath}\textnormal{Minimize } f_{2}(\vec{x}) =\sum^{n}_{i=1} \left( \vert x_{i}\vert^{0.8} + 5 \sin(x_{i})^{3} \right)\end{displaymath}$

(2)

where:

$\begin{displaymath}-5 \leq x_{1}, x_{2}, x_{3} \leq 5\end{displaymath}$

(3)

Pareto front.

Pareto front (data file)

DEB Bimodal

Bimodal problem proposed by Deb [1]:

$\begin{displaymath}\textnormal{Minimize } f_{1}(x_{1},x_{2}) = x_{1}\end{displaymath}$

(4)

$\begin{displaymath}\textnormal{Minimize } f_{2}(x_{1},x_{2}) = \frac{g(x_{2})}{x_{1}}\end{displaymath}$

(5)

$\begin{displaymath}g(x_{2})=2.0- \exp \left\{ - \left( \frac{x_{2}-0.2}{0.004} ......xp \left\{ - \left( \frac{x_{2}-0.6}{0.4} \right)^{2} \right\}\end{displaymath}$

(6)

and, $0.1 \leq x_{1} \leq 1.0$ , $0.1 \leq x_{2} \leq 1.0$ .

Pareto front.

Pareto front (data file)

Kita

Constraint problem proposed by Kita [3]:

$\begin{displaymath}\textnormal{Maximize } f_{1}(x,y) = - x^{2} + y\end{displaymath}$

(7)

$\begin{displaymath}\textnormal{Maximize } f_{2}(x,y) = \frac{1}{2}x + y + 1\end{displaymath}$

(8)

subject to:

$\begin{displaymath}\frac {1}{6}x + y - \frac{13}{2} \leq 0\end{displaymath}$

(9)

$\begin{displaymath}\frac {1}{2}x + y - \frac{15}{2} \leq 0\end{displaymath}$

(10)

$\begin{displaymath}\frac 5x + y - 30 \leq 0\end{displaymath}$

(11)

and, $0 \leq x \leq 7.0$ , $0 \leq y \leq 7.0$ .

Pareto front.

Pareto front (data file)

DTLZ1

Proposed by Deb et al [2]. It Requires 7 variables.

$\begin{displaymath}\begin{array}{cl}Minimize& f_1(\mathbf{x})={{1}\over{2}}x_1......ect to& 0 \leq xi \leq 1, for i =1,2,\ldots,n.\\\end{array}\end{displaymath}$

(12)

Pareto front.

Pareto front (data file)

DTLZ6

This problem was proposed by Deb et al [2]. This function uses 22 variables and has $2^{20}$ desconected regions.

$\begin{displaymath}\begin{array}{cl}Minimize &f_1(\mathbf{x})=x_1,\\\vdots&\......\subject to&0 \leq x_i \leq, for i=1,2..,n,\\\end{array}\end{displaymath}$

(13)

Pareto front

Pareto front (data file)

References

1: Kalyanmoy Deb.
Multi-Objective Genetic Algorithms: Problem Difficulties and Construction of Test Problems.
Evolutionary Computation, 7(3):205-230, Fall 1999.
2: Kalyanmoy Deb, Lothar Thiele, Marco Laumanns, and Eckart Zitzler.
Scalable Test Problems for Evolutionary Multi-Objective Optimization.
Technical Report 112, Computer Engineering and Networks Laboratory (TIK), Swiss Federal Institute of Technology (ETH), Zurich, Switzerland, 2001.
3: Hajime Kita, Yasuyuki Yabumoto, Naoki Mori, and Yoshikazu Nishikawa.
Multi-Objective Optimization by Means of the Thermodynamical Genetic Algorithm.
In Hans-Michael Voigt, Werner Ebeling, Ingo Rechenberg, and Hans-Paul Schwefel, editors, Parallel Problem Solving from Nature--PPSN IV, Lecture Notes in Computer Science, pages 504-512, Berlin, Germany, September 1996. Springer-Verlag.
4: Frank Kursawe.
A variant of evolution strategies for vector optimization.
In H. P. Schwefel and R. Männer, editors, Parallel Problem Solving from Nature. 1st Workshop, PPSN I, volume 496 of Lecture Notes in Computer Science, pages 193-197, Berlin, Germany, oct 1991. Springer-Verlag.

Gregorio Toscano Pulido 2003-07-01