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ETH Zürich - D-ITET - TIK - SOP - Downloads & Materials - Supplementary Materials - Testproblems - Zdt1
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Untitled Document

ZDT 1

Formulation:
Pareto Front:

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:

Optimal Distributions:


(see "Maximum Hypervolume" for more plots)
of 5 points:
of 10 points:
of 20 points:
of 50 points:
Maximum Hypervolume:
µHV ValuesPlot
2120.024876downloadplot
3120.387728downloadplot
4120.491597downloadplot
5120.539729downloadplot
10120.613761downloadplot
20120.642396downloadplot
50120.657446downloadplot
100120.662137downloadplot
1000120.666177downloadplot
120 2/3

How to approximate the optimal distributions

  1. The x-values xi of the µ points are equally distributed between 0 and 1
  2. The last point is known to be an extremal value, hence only the remaining µ -1 points are optimized.
  3. All points p with 1 <= p < µ, starting with point p = 1, are optimized according to the following formulas:

    For p = 1 (leftmost point)
    The value of x1 is set to aopt(r,b), where r is the y-value of the reference point and b is x2.

    For p > 1 and p < mu (non-extremal points)
    The value of xp is set to bopt(a,c), where a is xp-1 and b is xp+1

  4. The Hypervolume-Indicator is calculated. If the value did increase less than a predefined value eps, the distribution of x is returned. Otherwise, step 3 is repeated.
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