benchmarkfcns.hartmann6

benchmarkfcns.hartmann6(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 Hartmann N. 6 benchmark function. SCORES = hartmann6(X) computes the value of the Hartmann N. 6 function at point X. hartmann6 accepts a matrix of size M-by-6 and returns a vector SCORES of size M-by-1 in which each row contains the function value for the corresponding row of X. Properties:

  • Global minimum: approx -3.32237

  • Location of global minimum: approx (0.20169, 0.150011, 0.476874, 0.275332, 0.311652, 0.6573)

  • Number of dimensions: 6

  • Recommended domain: [0, 1]^6

  • Number of local minima: 6

  • Number of global minima: 1

  • Convexity: Non-convex

  • Separability: Non-separable

  • Modality: Multimodal

  • Symmetry: Non-symmetric

  • Differentiable: Yes

For more information, please visit: benchmarkfcns.info/doc/hartmann6fcn

Mathematical Definition

\[\begin{split}f(x) = -\sum_{i=1}^{4} \alpha_i \exp \left( -\sum_{j=1}^{6} A_{ij} (x_j - P_{ij})^2 \right) \\\end{split}\]

A = begin{bmatrix}10 & 3 & 17 & 3.5 & 1.7 & 10 \ 0.05 & 10 & 17 & 8 & 0.1 & 14 \ 3 & 3.5 & 1.7 & 10 & 17 & 8 \ 17 & 8 & 0.05 & 10 & 0.1 & 14end{bmatrix} \ P = begin{bmatrix}0.1312 & 0.1696 & 0.5569 & 0.0124 & 0.8283 & 0.5886 \ 0.2329 & 0.4135 & 0.8307 & 0.3736 & 0.1004 & 0.9991 \ 0.2348 & 0.1451 & 0.3522 & 0.2883 & 0.3047 & 0.6650 \ 0.4047 & 0.8828 & 0.8732 & 0.5743 & 0.1091 & 0.0381end{bmatrix} \ alpha = begin{bmatrix}1.0 & 1.2 & 3.0 & 3.2end{bmatrix}

Visualization

No visualization available for this function.