ETH - SOP - Downloads/Material - Supplementary Material - Testproblems

this webpage might no longer be updated more...

DTLZ 1

DTLZ 3

Formulation:

Pareto Front:

Relevant Publications:

K. Deb, L. Thiele, M. Laumanns and E. Zitzler. Scalable Multi-Objective Optimization Test Problems. CEC 2002, p. 825 - 830, IEEE Press, 2002 (PDF) (bibtex)

Reference Point:

Reference Point used:

Density:

Optimal Distributions:

(see "Maximum Hypervolume" for more plots)

of 5 points:

of 10 points:

of 20 points:

of 50 points:

Maximum Hypervolume:

µ	HV	Values	Plot
2	120.000000	download	plot
3	120.085786	download	plot
4	120.121585	download	plot
5	120.141536	download	plot
10	120.178966	download	plot
20	120.196858	download	plot
50	120.207485	download	plot
100	120.210643	download	plot
1000	120.214114	download	plot
∞	121-1/4 π

The x-values x_i of the µ points are equally distributed between 0 and 1
All µ points are optimized.
For all points p, starting with point p = 1, the following steps are executed:
- The x-value is decreased by stepsize. If this increases the hypervolume, the procedure continues with the next point.
- If decreasing x_p does not increase the hypervolume, then x_p is increased by 2*stepsize. If this does not increase the hypervolume, x_p decreased by stepsize (which means it has the value at the start of step 3).
If step 3 did increase the hypervolume, step 3 is repeated as long as the hypervolume increases. Otherwise, step 5 is executed
If step 5 did not increase the hypervolume, stepsize is scaled down by 1/2. If stepsize is smaller than a predefined value eps (10^-16), the precedure returns the current distribution. Otherwise, the precedure restarts with step 3.