Alpine N. 1 Function
Mathematical Definition
\[f(\mathbf x)=f(x_1, ..., x_n) = \sum_{i=1}^{n}|x_i sin(x_i)+0.1x_i|\]Plots
Two contours of the function are presented below:
Description and Features
- The function is not convex.
- The function is defined on n-dimensional space.
- The function is non-separable.
- The function is differentiable.
Input Domain
The function can be defined on any positive input domain but it is usually evaluated on $x_i \in [0, 10]$ for $i=1, …, n$.
Global Minima
The function has a global minimum $f(\textbf{x}^{\ast})=0$ located at $\mathbf{x^\ast}=(0, …, 0)$.
Implementation
Python
For Python, the function is implemented in the benchmarkfcns package and can be installed from command line with pip install benchmarkfcns.
from benchmarkfcns import alpine1
print(alpine1([[0, 0, 0],
[1, 1, 1]]))MATLAB
An implementation of the Alpine N. 1 Function with MATLAB is provided below.
% Computes the value of the Alpine N. 1 function.
% SCORES = ALPINEN1FCN(X) computes the value of the Alpine N. 1
% function at point X. ALPINEN1FCN accepts a matrix of size M-by-N and
% returns a vetor SCORES of size M-by-1 in which each row contains the
% function value for the corresponding row of X.
%
% Author: Mazhar Ansari Ardeh
% Please forward any comments or bug reports to mazhar.ansari.ardeh at
% Google's e-mail service or feel free to kindly modify the repository.
function scores = alpinen1fcn(x)
scores = sum(abs(x .* sin(x) + 0.1 * x), 2);
end The function can be represented in Latex as follows:
f(\mathbf x)=f(x_1, ..., x_n) = \sum{i=1}^{n}|x_i sin(x_i)+0.1x_i|See Also:
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
- M. Clerc, “The Swarm and the Queen, Towards a Deterministic and Adaptive Particle Swarm Optimization, ” IEEE Congress on Evolutionary Computation, Washington DC, USA, pp. 1951-1957, 1999.