benchmarkfcns.helicalvalley

benchmarkfcns.helicalvalley(arg0: Annotated[numpy.typing.NDArray[numpy.float64], '[m, n]', 'flags.c_contiguous']) → Annotated[numpy.typing.NDArray[numpy.float64], '[m, 1]']

Computes the value of the Helical Valley benchmark function. SCORES = helicalvalley(X) computes the value of the function at point X. helicalvalley accepts a matrix of size M-by-3 and returns a vector SCORES of size M-by-1. Properties:

• Global minimum: 0

• Location of global minimum: (1, 0, 0)

• Number of dimensions: 3

• Recommended domain: [-10, 10]^3

• Modality: unimodal

Mathematical Definition

\[f(x_1, x_2, x_3) = 100 \left[ (x_3 - 10\theta(x_1, x_2))^2 + (\sqrt{x_1^2 + x_2^2} - 1)^2 \right] + x_3^2\]

text{where } theta(x_1, x_2) = frac{1}{2pi} operatorname{atan2}(x_2, x_1)

Visualization

No visualization available for this function.