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ETH Zürich - D-ITET - TIK - SOP - Downloads & Materials - Supplementary Materials - Testproblems - Zdt3
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Untitled Document

ZDT 3

Formulation:
Pareto Front:

with and where

Relevant Publications:
  • E. Zitzler, K. Deb, and L. Thiele. Comparison of Multiobjective Evolutionary Algorithms: Empirical Results. Evolutionary Computation, 8(2):173-195, 2000 (PDF) (bibtex)
Reference Point: Reference Point used: (11,11)
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:
µHV ValuesPlot
2127.996445downloadplot
3128.448485downloadplot
4128.598233downloadplot
5128.665669downloadplot
10128.748470downloadplot
20128.765657downloadplot
50128.773637downloadplot
100128.775955downloadplot
1000128.777722downloadplot
128.77811613069076060

How to approximate the optimal distributions

  1. The x-values xi of the µ points are equally distributed between 0 and 0.851832865436414
  2. The first and last point are known to be extremal, hence only the remaining µ -2 points are optimized.
  3. 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 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. For all points p, xp is set to a random value between 0 and 0.851832865436414 until the hypervolume increases or for maximum 10 times. This allows points to jump to different sections of the Pareto front.
  6. If the hypervolume increased in step 5, the procedure starts all over again with step 3.
  7. 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.
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