KURSAWE
Proposed by Kursawe  [4].; $P_{true}$ disconnected, $Pf_{true}$disconnected	Minimize $F = (f_1(\vec{x}), f_2(\vec{x}))$, where $$f_1(\vec{x}) = \sum\limits_{i=1}^{2}-10e^{\text{-0.2} \sqrt{x_{i}^{2}+x_{i+1}^{2}}},$$ $$f_2(\vec{x}) = \sum\limits_{i=1}^{2}\left( \left\vert x_{i}\right\vert ^\text{0.8}+5\sin (x_{i}^{3})\right)$$	$$-5 \leq x_{i} \leq 5,$$ $$i = 1,2, 3$$
Kursawe Pareto Front (data file)

DEB Bimodal
Proposed by Deb [1].; $P_{true}$connected, $Pf_{true}$connected	Minimize $F = (f_1(\vec{x}), f_2(\vec{x}))$, where $$f_1(\vec{x}) = x_1,$$ $$f_2(\vec{x}) = \frac{g(\vec{x})}{x_1},$$ $$g(\vec{x}) = 2.0 - e^{-\left(\frac{x_2 - 0.2}{0.004}\right)^2} - 0.8 \cdot e^{-\left(\frac{x_2 - 0.6}{0.4}\right)^2}$$	$$0.1 \leq x_{i} \leq 1,$$ $$i = 1,2$$
Deb Pareto Front (data file)

KITA
Proposed by Kita  [3]; $P_{true}$connected, $Pf_{true}$concave; Nondominated solutions are in the border of Feasible region	Maximize $F = (f_1(\vec{x}), f_2(\vec{x}))$, where $$f_1(\vec{x}) = - x_1^2 + x_2,$$ $$f_2(\vec{x}) = \frac{1}{2}x_1 + x_2 + 1,$$	subject to: $$\frac{1}{6}x_1 + x_2 - \frac{13}{2} \leq$$ 0 $$\frac{1}{2}x_1 + x_2 - \frac{15}{2} \leq$$ 0 $$\frac{5}{x_1} + x_2 - 30 \leq$$ 0 $$0 \leq x_{i} \leq 7$$ $$i = 1, 2$$
KITA Pareto Front (data file)

DTLZ1
Proposed by Deb et. al [2]	Minimize $F = (f_1(\vec{x}), f_2(\vec{x}), f_3(\vec{x}))$, where $$f_{1}(\vec{x}) = \frac{1}{2}x_{1}x_{2}(1+g(\vec{x})),$$ $$f_{2}(\vec{x}) = \frac{1}{2}x_{1}(1-x_{2})(1+g(\vec{x})),$$ $$f_{2}(\vec{x}) = \frac{1}{2}(1-x_{1})(1+g(\vec{x})).$$ $$and$$ $$g(\vec{x}) = 100[10 + \sum\limits_{i=3}^{n}\left(x_i-0.5\right)^{2}$$ $$- \cos(20\pi(x_i - 0.5))].$$	$$n = 12,$$ $$0\leq x_{i} \leq 1,$$ $$i = 1,...,12$$
DTLZ1 Pareto Front (data file)

DTLZ7
Proposed by Deb et. al [2]	Minimize $F = (f_1(\vec{x}), f_2(\vec{x}), f_3(\vec{x}))$, where $$f_{1}(\vec{x}) = x_{1}$$ $$f_{2}(\vec{x}) = x_{2}$$ $$f_{3}(\vec{x}) = (1 + g(\vec{x})) \cdot h(f_1, f_2, g(\vec{x}))$$ $$and:$$ $$g(\vec{x}) = 1 + \frac{9}{22} \cdot \sum\limits_{i=3}^{n}\left( x_i\right)$$ $$h(f_1, f_2, g) = 3- \sum\limits_{i=1}^{2}\left( \frac{f_i}{1+g} (1 + sin(3\pi f_i))\right)$$	$$n = 22,$$ $$0\leq x_{i} \leq 1,$$ $$i = 1,...,12$$
DTLZ7 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.