Mathematical Definition

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

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