makeHartmannFunction function

Hartmann Function

Unimodal single-objective test function with six local minima. The implementation is based on the mathematical formulation [REMOVE_ME]$f(x) = - \sum_{i=1}^4 \alpha_i \ exp \left(-\sum_{j=1}^6 A_{ij}(x_j-P_{ij})^2 \right) [REMOVE_ME_2]$, where [REMOVE_ME]$\alpha = (1.0, 1.2, 3.0, 3.2)^T, \\A = \left( \begin{array}{rrrrrr}10 & 3 & 17 & 3.50 & 1.7 & 8 \\0.05 & 10 & 17 & 0.1 & 8 & 14 \\3 & 3.5 & 1.7 & 10 & 17 & 8 \\17 & 8 & 0.05 & 10 & 0.1 & 14\end{array} \right), \\P = 10^{-4} \cdot \left(\begin{array}{rrrrrr}1312 & 1696 & 5569 & 124 & 8283 & 5886 \\2329 & 4135 & 8307 & 3736 & 1004 & 9991 \\2348 & 1451 & 3522 & 2883 & 3047 & 6650 \\4047 & 8828 & 8732 & 5743 & 1091 & 381\end{array} \right) [REMOVE_ME_2]$

The function is restricted to six dimensions with $\mathbf{x}_i \in [0,1], i = 1, \ldots, 6.$

The function is not normalized in contrast to some benchmark applications in the literature.

makeHartmannFunction(dimensions)

Arguments

• dimensions: [integer(1)]

  Size of corresponding parameter space.

Returns

[smoof_single_objective_function]

Description

Unimodal single-objective test function with six local minima. The implementation is based on the mathematical formulation

, where

The function is restricted to six dimensions with $\mathbf{x}_i \in [0,1], i = 1, \ldots, 6.$

The function is not normalized in contrast to some benchmark applications in the literature.

References

Picheny, V., Wagner, T., & Ginsbourger, D. (2012). A benchmark of kriging-based infill criteria for noisy optimization.