Mathematical Definition

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Cross-in-Tray Function

Cross-in-Tray Function

Cross-in-Tray Function

Cross-in-Tray Function

The contour of the function is as presented below:

Cross-in-Tray 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 non-differentiable.
  • The function is .
  • The function is .

Input Domain

The function can be defined on any input domain but it is usually evaluated on $x \in [-10, 10]$ and $y \in [-10, 10]$ .

Global Minima

The function has four global minima $f(\textbf{x}^{\ast})=-2.06261218$ at $\textbf{x}^{\ast} = (\pm1.349406685353340,\pm1.349406608602084)$.

Implementation

An implementation of the Cross-in-Tray Function with MATLAB is provided below.

% Computes the value of the Cross-in-tray benchmark function.
% SCORES = CROSSINTRAYFCN(X) computes the value of the Cross-in-tray 
% function at point X. CROSSINTRAYFCN 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 = crossintrayfcn(x)
    
    n = size(x, 2);
    assert(n == 2, 'The Cross-in-tray function is only defined on a 2D space.')
    X = x(:, 1);
    Y = x(:, 2);

    expcomponent = abs(100 - (sqrt(X .^2 + Y .^2) / pi));
    
    scores = -0.0001 * ((abs(sin(X) .* sin(Y) .* exp(expcomponent)) + 1) .^ 0.1);
end

The function can be represented in Latex as follows:

f(x,y)=-0.0001(|sin(x)sin(y)exp(|100-\frac{\sqrt{x^2+y^2}}{\pi}|)|+1)^{0.1}

References: