Add the Eight-Dimensional Function from Dette and Pepelyshev (2010)

[damar-wicaksono]

The eight-dimensional function from Dette and Pepelyshev (2010)¹ is a test function used in the context of metamodeling (specifically, design of computer experiments to construct metamodels). The function reads as follows:

$$ \mathcal{M}(\boldsymbol{x}) = 4 \left( x_1 - 2 + 8 x_2 - 8 x_2^2 \right) + \left( 3 - 4 x_2 \right)^2 + 16 \left( x_3 + 1 \right)^{0.5} \left( 2 x_3 - 1 \right)^2 + \sum_{k = 4}^8 k \ln{\left( 1 + \sum_{i=3}^k x_i \right)} $$

where $\boldsymbol{x} = \{ x_1, \ldots, x_8 \}$ is the vector of input variables modeled as an independent uniform random variable in $[0, 1]^8$. The function is highly curved with respect to some input variables and less so with respect to some others.

Footnotes

1. H. Dette and A. Pepelyshev, "Generalized latin hypercube design for computer experiments," Technometrics, vol. 52, no. 4, pp. 421-429, 2010. doi: 10.1198/TECH.2010.09157. See Eq. (6), Section 3.3. ↩