benchmarkfcns.hartmann3

benchmarkfcns.hartmann3(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. 3 benchmark function. SCORES = hartmann3(X) computes the value of the Hartmann N. 3 function at point X. hartmann3 accepts a matrix of size M-by-3 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 -3.86278

  • Location of global minimum: approx (0.114614, 0.555649, 0.852547)

  • Number of dimensions: 3

  • Recommended domain: [0, 1]^3

  • Number of local minima: 4

  • 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/hartmann3fcn

Mathematical Definition

\[\begin{split}f(x) = -\sum_{i=1}^{4} \alpha_i \exp \left( -\sum_{j=1}^{3} A_{ij} (x_j - P_{ij})^2 \right) \\\end{split}\]

A = begin{bmatrix}3 & 10 & 30 \ 0.1 & 10 & 35 \ 3 & 10 & 30 \ 0.1 & 10 & 35end{bmatrix} \ P = begin{bmatrix}0.3689 & 0.1170 & 0.2673 \ 0.4699 & 0.4387 & 0.7470 \ 0.1091 & 0.8732 & 0.5547 \ 0.0381 & 0.5743 & 0.8828end{bmatrix} \ alpha = begin{bmatrix}1.0 & 1.2 & 3.0 & 3.2end{bmatrix}

Visualization

No visualization available for this function.