Test Cases

A two-dimensional Branin function is tested for the one-shot method, and a three-dimensional Hartmann function is tested for the sequential sampling method. In all cases, the Halton sequence is applied to ensure consistent starting points for all methods.

One-shot method - Branin function

The minimization of the two-dimensional Branin function is formulated as:

\[\min_{x_{d_1}, \text{ } x_{d_2}} f(x_{d_1},x_{d_2}) = \left[ x_{d_2} - \frac{5.1}{4\pi^2}x_{d_1}^2 + \frac{5}{\pi}x_{d_1} - 6 \right]^2 + 10\left[ 1 - \frac{1}{8\pi} \right] \cos(x_{d_1}) + 10,\]

with constraints \(x_{d1} \in [0, 8]\) and \(x_{d2} \in [0, 8]\). The goal is to find a global minimum of 0.39 at \((\pi, 2.275)\).

Sequential sampling method - Hartmann function

The minimization of the three-dimensional Hartmann function is expressed as:

\[\min_{\mathbf{x}_d} f(\mathbf{x}_d) = -\sum_{i=1}^{4} \alpha_i \exp \left( -\sum_{j=1}^{3} A_{ij}(x_{dj} - P_{ij})^2 \right),\]

where \(\alpha = [1.0, 1.2, 3.0, 3.2]^T\), and matrices \(\mathbf{A}\) and \(\mathbf{P}\) are defined as:

\[\begin{split}\mathbf{A} = \begin{bmatrix} 3.0 & 10 & 30 \\ 0.1 & 10 & 35 \\ 3.0 & 10 & 30 \\ 0.1 & 10 & 35 \end{bmatrix}, \quad \mathbf{P} = 10^{-4} \begin{bmatrix} 3689 & 1170 & 2673 \\ 4699 & 4387 & 7470 \\ 1091 & 8732 & 5547 \\ 381 & 5743 & 8828 \end{bmatrix}.\end{split}\]

The function is constrained by \(\mathbf{x}_{d_L} = (0,0,0)\) and \(\mathbf{x}_{d_U} = (1,1,1)\), with one global minimum of -3.86 at \((0.11, 0.56, 0.85)\) and four local minima.