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ETH Zürich - D-ITET - TIK - SOP - Downloads & Materials - Supplementary Materials - Testproblems - Dtlz2
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DTLZ 1

DTLZ 2

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: (11,11)
Density:

Optimal Distributions:

(see "Maximum Hypervolume" for more plots)

of 5 points:
of 10 points:
of 20 points:
of 50 points:
Maximum Hypervolume:
µHV ValuesPlot
2120.000000downloadplot
3120.085786downloadplot
4120.121585downloadplot
5120.141536downloadplot
10120.178966downloadplot
20120.196858downloadplot
50120.207485downloadplot
100120.210644downloadplot
1000120.214114downloadplot
121-1/4 π

How to approximate the optimal distributions

  1. The x-values xi of the µ points are equally distributed between 0 and 1
  2. All µ points are optimized.
  3. 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 xp does not increase the hypervolume, then xp is increased by 2*stepsize. If this does not increase the hypervolume, xp decreased by stepsize (which means it has the value at the start of step 3).


  4. If step 3 did increase the hypervolume, step 3 is repeated as long as the hypervolume increases. Otherwise, step 5 is executed
  5. 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.
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