Mathematical Definition

Plots

Beale Function

The contour of the function is as presented below:

Beale Function

Description and Features

  • The function is continuous.
  • The function is not convex.
  • The function is defined on 2-dimensional space.
  • The function is multimodal.

Input Domain

The function can be defined on any input domain but it is usually evaluated on $x \in [-4.5, 4.5]$ for all $i = 1, 2$.

Global Minima

The function has one global minimum at: $f(x^*)=0$ at $\textbf{x}^{\ast} = (3, 0.5)$.

Implementation

An implementation of the Beale Function with MATLAB is provided below.

% Computes the value of the Beale benchmark function.
% SCORES = BEALEFCN(X) computes the value of the Beale function at 
% point X. BEALEFCN accepts a matrix of size M-by-2 and returns a  
% vetor SCORES of size M-by-1 in which each row contains the function value 
% for the corresponding 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 scores = bealefcn(x)
    n = size(x, 2);
    assert(n == 2, 'Beale''s function is only defined on a 2D space.')
    X = x(:, 1);
    Y = x(:, 2);
    
    scores = (1.5 - X + (X .* Y)).^2 + ...
             (2.25 - X + (X .* (Y.^2))).^2 + ...
             (2.625 - X + (X .* (Y.^3))).^2;
end

The function can be represented in Latex as follows:

f(x, y) = (1.5-x+xy)^2+(2.25-x+xy^2)^2+(2.625-x+xy^3)^2

References: