benchmarkfcns.multiobjective.zdt4

benchmarkfcns.multiobjective.zdt4(arg0: Annotated[numpy.typing.NDArray[numpy.float64], '[m, n]', 'flags.c_contiguous']) → Annotated[numpy.typing.NDArray[numpy.float64], '[m, n]']

Computes the value of the ZDT4 multi-objective benchmark function. SCORES = multiobjective.zdt4(X) computes the value of the ZDT4 function at point X. multiobjective.zdt4 accepts a matrix of size M-by-N and returns a matrix SCORES of size M-by-2. Properties:

• Global Pareto front: f_2 = 1 - sqrt{f_1}

• Number of dimensions: n (usually 10)

• Recommended domain: x_1 ∈ [0, 1], x_i ∈ [-5, 5] for i > 1

• Modality: Highly Multimodal (has 21^{n-1} local Pareto fronts)

For more information, please visit: benchmarkfcns.info/doc/zdt4fcn

Mathematical Definition

\[\begin{split}f_1(\textbf{x}) = x_1 \\\end{split}\]

f_2(textbf{x}) = g(textbf{x}) h(f_1, g) \ g(textbf{x}) = 1 + 10(n-1) + sum_{i=2}^{n} (x_i^2 - 10 cos(4pi x_i)) \ h(f_1, g) = 1 - sqrt{frac{f_1}{g}}

Visualization

No visualization available for this function.