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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:

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	120.024876	download	plot
3	120.387728	download	plot
4	120.491597	download	plot
5	120.539729	download	plot
10	120.613761	download	plot
20	120.642396	download	plot
50	120.657446	download	plot
100	120.662137	download	plot
1000	120.666177	download	plot
∞	120 2/3

The x-values x_i of the µ points are equally distributed between 0 and 1
The last point is known to be an extremal value, hence only the remaining µ -1 points are optimized.
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 a_opt(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 x_p is set to bopt(a,c), where a is x_p-1 and b is x_p+1
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.