Mathematical Definition

Plots

Keane Function

Keane Function

Keane Function

Keane Function

Contour of the function is presented below:

Keane 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.
  • The function is differentiable.
  • The function is non-separable.

Input Domain

The function can be defined on any input domain but it is usually evaluated on $x_i \in [0, 10]$ for $i=1, 2$.

Global Minima

The function has two global minima $f(\textbf{x}^{\ast})=0.673667521146855$ at

  • $\textbf{x}^{\ast} = (1.393249070031784,0)$.
  • $\textbf{x}^{\ast} = (0,1.393249070031784)$.

Implementation

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

% Computes the value of the Keane function.
% SCORES = KEANEFCN(X) computes the value of the Keane function at point X.
% KEANEFCN 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.
% 
% 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 = keanefcn(x)
    n = size(x, 2);
    assert(n == 2, 'Keane function is defined only on a 2D space.')
    X = x(:, 1);
    Y = x(:, 2);
    
    numeratorcomp = (sin(X - Y) .^ 2) .* (sin(X + Y) .^ 2); 
    denominatorcomp = sqrt(X .^2 + Y .^2);
    scores = - numeratorcomp ./ denominatorcomp;
end

The function can be represented in Latex as follows:

f(x,y)=-\frac{\sin^2(x-y)\sin^2(x+y)}{\sqrt{x^2+y^2}}

References: