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DTLZ 1

DTLZ 7

Formulation:

Pareto Front:

with and where

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:

where (see Formulation)

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	112.534293	download	plot
3	114.676857	download	plot
4	115.495800	download	plot
5	115.748424	download	plot
10	115.964708	download	plot
20	116.045710	download	plot
50	116.089454	download	plot
100	116.101551	download	plot
1000	116.111529	download	plot
∞	116.1138716447221

The x-values x_i of the µ points are equally distributed between 0 and 2.116426807
The first and last point are known to be extremal, hence only the remaining µ -2 points are optimized.
For all points p, starting with point p = 2, 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
For all points p, x_p is set to a random value between 0 and 2.116426807 until the hypervolume increases or for maximum 10 times. This allows points to jump to different sections of the Pareto front.
If the hypervolume increased in step 5, the procedure starts all over again with step 3.
If no jump in step 5 did 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.