Mathematical Definition

\[f(x,y)=100\sqrt{|y-0.01x^2|}+0.01|x+10|\]

Plots

Bukin N. 6 Function

Bukin N. 6 Function

Bukin N. 6 Function

The contour of the function is as presented below:

Bukin N. 6 Function

Description and Features

  • The function is continuous.
  • The function is 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 non-scalable.

Input Domain

The function can be defined on any input domain but it is usually evaluated on $x \in [-15, -5]$ and $y \in [-3, 3]$ .

Global Minima

The function has one global minimum at: $f(\textbf{x}^{\ast})=0$ at $\textbf{x}^{\ast} = (-10,1)$.

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 bukin6

print(bukin6([[0, 0, 0],
              [1, 1, 1]]))

MATLAB

An implementation of the Bukin N. 6 Function with MATLAB is provided below.

% Computes the value of the Bukin N. 6 benchmark function.
% SCORES = BUKINN6FCN(X) computes the value of the Bukin N. 6 function at 
% point X. BUKINN6FCN 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 = bukinn6fcn(x)
    n = size(x, 2);
    assert(n == 2, 'The Bukin N. 6 functions is only defined on a 2D space.')
    
    X = x(:, 1);
    X2 = X .^ 2;
    Y = x(:, 2);
    
    scores = 100 * sqrt(abs(Y - 0.01 * X2)) + 0.01 * abs(X  + 10);
end

The function can be represented in Latex as follows:

f(x,y)=100\sqrt{|y-0.01x^2|}+0.01|x+10|

References: