Unimodal single-objective test function with six local minima. The implementation is based on the mathematical formulation $$f(x) = - \sum_{i=1}^4 \alpha_i \ exp \left(-\sum_{j=1}^6 A_{ij}(x_j-P_{ij})^2 \right)$$, where $$\alpha = (1.0, 1.2, 3.0, 3.2)^T, \\ A = \left( \begin{array}{rrrrrr} 10 & 3 & 17 & 3.50 & 1.7 & 8 \\ 0.05 & 10 & 17 & 0.1 & 8 & 14 \\ 3 & 3.5 & 1.7 & 10 & 17 & 8 \\ 17 & 8 & 0.05 & 10 & 0.1 & 14 \end{array} \right), \\ P = 10^{-4} \cdot \left(\begin{array}{rrrrrr} 1312 & 1696 & 5569 & 124 & 8283 & 5886 \\ 2329 & 4135 & 8307 & 3736 & 1004 & 9991 \\ 2348 & 1451 & 3522 & 2883 & 3047 & 6650 \\ 4047 & 8828 & 8732 & 5743 & 1091 & 381 \end{array} \right)$$ 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)[integer(1)]
Size of corresponding parameter space.
[smoof_single_objective_function]
Picheny, V., Wagner, T., & Ginsbourger, D. (2012). A benchmark of kriging-based infill criteria for noisy optimization.