Adjustable Hartmann3

Implementation of the adjustable bi-fidelity Hartmann3 function as defined in:

Toal, D.J.J. Some considerations regarding the use of multi- fidelity Kriging in the construction of surrogate models. Struct Multidisc Optim 51, 1223–1245 (2015) doi:10.1007/s00158-014-1209-5

Function definitions:

\[f_h(x_1, x_2, x_3) = -\sum^4_{i=1} \alpha_i \exp \Bigg(-\sum^3_{j=1} \beta_{ij} (x_j - P_{ij})^2 \Bigg)\]

\[f_l(x_1, x_2, x_3) = -\sum^4_{i=1} \alpha_i \exp \Bigg(-\sum^3_{j=1} \beta_{ij} \Big(x_j - \dfrac{3}{4}P_{ij}(a + 1)\Big)^2 \Bigg)\]

with the following matrices and vectors:

\[\begin{split}\beta = \left( \begin{array}{cccccc} 3 & 10 & 30 \\ 0.1 & 10 & 35 \\ 3 & 10 & 30 \\ 0.1 & 10 & 35 \end{array} \right)\end{split}\]

\[\begin{split}P = 10^{-4} \left( \begin{array}{cccccc} 3689 & 1170 & 2673 \\ 4699 & 4387 & 7470 \\ 1091 & 8732 & 5547 \\ 381 & 5743 & 8828 \end{array} \right)\end{split}\]

\[\alpha = \{1.0, 1.2, 3.0, 3.2\}\]

and where \(a \in [0, 1]\) is the adjustable parameter.

adjustable_hartmann3_lf(xx, a)

hartmann3_hf(xx)