The eight-dimensional function from Dette and Pepelyshev (2010)[^dette] is a test function used in the context of metamodeling (specifically, design of computer experiments to construct metamodels). The function reads as follows:
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.
[^dette]: 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.
The eight-dimensional function from Dette and Pepelyshev (2010)[^dette] is a test function used in the context of metamodeling (specifically, design of computer experiments to construct metamodels). The function reads as follows:
$$ \mathcal{M}(\boldsymbol{x}) = 4 \left( x_1 - 2 + 8 x_2 - 8 x_2^2 \right) + \left( 3 - 4 x_2 \right)^2 + 16 \left( x_3 + 1 \right)^{0.5} \left( 2 x3 - 1 \right)^2 + \sum{k = 4}^8 k \ln{\left( 1 + \sum_{i=3}^k x_i \right)} $$
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.
[^dette]: 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.