Sphere Function
Mathematical Definition
\[f(\textbf{x}) = f(x_1, x_2, ..., x_n) = {\sum_{i=1}^{n} x_i^{2}}.\]Plots
The contour of the function:
Description and Features
- The function is continuous.
- The function is convex.
- The function can be defined on n-dimensional space.
- The function is differentiable.
- The function is separable.
- The function is unimodal.
Input Domain
The function can be defined on any input domain but it is usually evaluated on the hypercube $x_i \in [-5.12, 5.12]$ for $i = 1..n$.
Global Minima
$f(\textbf{x}^{\ast}) = 0$ at $\textbf{x}^{\ast} = (0, …, 0)$
Implementation
Python
For Python, the function is implemented in the benchmarkfcns package, which can be installed from command line with pip install benchmarkfcns
.
from benchmarkfcns import sphere
print(sphere([[0, 0, 0],
[1, 1, 1]]))
MATLAB
An implementation of the Sphere function with MATLAB is provided below. Sphere function can be implemented with a for
loop that iterates over all the components of the input vector but, MATLAB and Octave have built-in facilities that makes the implementation more efficient and concise.
% Computes the value of Sphere benchmark function.
% SCORES = SPHEREFCN(X) computes the value of the Ackey function at
% point X. SPHEREFCN 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
% each row of X.
% For more information please visit:
% https://en.wikipedia.org/wiki/Test_functions_for_optimization
%
% 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 f = spherefcn(x)
f = sum(x .^ 2, 2);
end
The function can be represented in Latex as follows:
f(\textbf{x}) = f(x_1, x_2, ..., x_n) = {\sum_{i=1}^{n} x_i^{2}}
References:
- http://www.sfu.ca/~ssurjano/spheref.html
- https://en.wikipedia.org/wiki/Test_functions_for_optimization