Mathematical Definition

\[f(\textbf{x}) = f(x_1, x_2, ..., x_n) = {\sum_{i=1}^{n} x_i^{2}}.\]

Plots

Sphere Function

The contour of the function: Sphere Function Contour

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