Mathematical Definition

\[f(x,y)=-|sin(x)cos(y)exp(|1-\frac{\sqrt{x^2+y^2}}{\pi}|)|\]

Plots

Holder-Table Function

Holder-Table Function

Holder-Table Function

The contour of the function is as presented below:

Holder-Table 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 non-separable.
  • 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 $f(\textbf{x}^{\ast})=-19.2085$ at $\textbf{x}^{\ast} = (\pm 8.05502,\pm 9.66459)$.

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 holdertable

print(holdertable([[0, 0],
              [1, 1]]))

MATLAB

An implementation of the Holder-Table Function with MATLAB is provided below.

% Computes the value of the Holder table benchmark function.
% SCORES = HOLDERTABLEFCN(X) computes the value of the Holder table  
% function at point X. HOLDERTABLEFCN 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 = holdertablefcn(x)
    
    n = size(x, 2);
    assert(n == 2, 'The Holder-table function is only defined on a 2D space.')
    X = x(:, 1);
    Y = x(:, 2);
    
    expcomponent = exp( abs(1 - (sqrt(X .^2 + Y .^ 2) / pi)) );
    
    scores = -abs(sin(X) .* cos(Y) .* expcomponent);
end

The function can be represented in Latex as follows:

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

References: