makeHartmannFunction: Hartmann Function

In smoof: Single and Multi-Objective Optimization Test Functions

Hartmann Function

Description

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.

Usage

makeHartmannFunction(dimensions)

Arguments

dimensions

[integer(1)] Size of corresponding parameter space.

Value

[smoof_single_objective_function]

References

Picheny, V., Wagner, T., & Ginsbourger, D. (2012). A benchmark of kriging-based infill criteria for noisy optimization.

smoof documentation built on March 31, 2023, 11:48 p.m.