Mathematical Definition

\[f(x, y) = 100(y − x^{3})^2 + (1 − x)^2\]

Plots

Leon Function

Leon Function

Leon Function

Leon Function

Contour of the function is presented below:

Leon Function

Description and Features

  • The function is continuous.
  • The function is not convex.
  • The function is defined on 2-dimensional space.
  • The function is unimodal.
  • 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 one global minimum $f(\textbf{x}^{\ast})=0$ at $\textbf{x}^{\ast} = (1,1)$.

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 leon

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

MATLAB

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

% Computes the value of the Leon function.
% SCORES = LEONFCN(X) computes the value of the Leon function at point X.
% LEONFCN 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 = leonfcn(x)
    n = size(x, 2);
    assert(n == 2, 'Leon function is defined only on a 2D space.')
    X = x(:, 1);
    Y = x(:, 2);
    
    scores = 100 * ((Y - X.^3) .^2) + ((1 - X) .^2);
end

The function can be represented in Latex as follows:

f(x, y) = 100(y − x^{3})^2 + (1 − x)^2

References: