Shubert Function
Mathematical Definition
\[f(\mathbf{x})=f(x_1, ...,x_n)=\prod_{i=1}^{n}{\left(\sum_{j=1}^5{ cos((j+1)x_i+j)}\right)}\]Plots
Two contours of the function are presented below:
Description and Features
- The function is continuous.
- The function is not convex.
- The function is defined on n-dimensional space.
- The function is multimodal.
- The function is differentiable.
- The function is non-separable.
Input Domain
The function can be defined on any input domain but it is usually evaluated on $x_i \in [-10, 10]$ for $i=1, …, n$.
Global Minima
The function has 18 global minima $f(\textbf{x}^{\ast})\approx-186.7309$.
Implementation
Python
For Python, the function is implemented in the benchmarkfcns package, which can be installed from command line with pip install benchmarkfcns
.
MATLAB
An implementation of the Shubert Function with MATLAB
is provided below.
The function can be represented in Latex as follows:
References:
- Momin Jamil and Xin-She Yang, A literature survey of benchmark functions for global optimization problems, Int. Journal of Mathematical Modelling and Numerical Optimisation}, Vol. 4, No. 2, pp. 150–194 (2013), arXiv:1308.4008
- E. P. Adorio, U. P. Dilman, “MVF - Multivariate Test Function Library in C for Unconstrained Global Optimization Methods,” [Available Online]: http://www.geocities.ws/eadorio/mvf.pdf