Mathematical Definition

\[f(x,y)=−cos(x)cos(y) exp(−(x − \pi)^2−(y − \pi)^2)\]

Plots

Easom Function

Easom Function

Easom Function

Easom Function

The contour of the function is as presented below:

Easom 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.
  • The function is non-scalable.

Input Domain

The function can be defined on any input domain but it is usually evaluated on $x \in [-100, 100]$ and $y \in [-100, 100]$ .

Global Minima

The function has four global minima $f(x^{\ast}, y^{\ast})=-1$ at $(x^{\ast}, y^{\ast}) = (\pi,\pi)$.

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 easom

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

MATLAB

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

% Computes the value of the Easom benchmark function.
% SCORES = EASOMFCN(X) computes the value of the Easom function at point X.
% EASOMFCN 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 = easomfcn(x)
    
    n = size(x, 2);
    assert(n == 2, 'The Easom''s function is only defined on a 2D space.')
    X = x(:, 1);
    Y = x(:, 2);
    
    scores = -cos(X) .* cos(Y) .* exp(-( ((X - pi) .^2) + ((Y - pi) .^ 2)) );
end

The function can be represented in Latex as follows:

f(x,y)=−cos(x)cos(y) exp(−(x − \pi)^2−(y − \pi)^2)

Acknowledgement

Tobias Völk kindly contributed to the correctness of this document.

References: