Mathematical Definition

\[f(x, y) = sin(x)e^{(1-cos(y))^2}+cos(y)e^{(1-sin(x))^2}+(x-y)^2\]

Plots

Two contours of the function are presented below:

Description and Features

• The function is not convex.
• The function is defined on 2-dimensional space.
• The function is non-separable.
• The function is differentiable.

Input Domain

The function can be defined on any input domain but it is usually evaluated on $x_i \in [-2\pi, 2\pi]$ for $i=1, 2$.

Global Minima

The function has two global minima at $f(\textbf{x}^{\ast}) = -106.764537$ located at $\mathbf{x^\ast}=(4.70104, 3.15294)$ and $\mathbf{x^\ast}=(-1.58214, -3.13024)$.

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 bird

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

MATLAB

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

% Computes the value of the Bird function.
% SCORES = BIRDFCN(X) computes the value of the Bird 
% function at point X. BIRDFCN 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 = birdfcn(x)
    
    n = size(x, 2);
    assert(n == 2, 'Bird function is only defined on a 2D space.')
    X = x(:, 1);
    Y = x(:, 2);
    
    scores = sin(X) .* exp((1 - cos(Y)).^2) + ... 
        cos(Y) .* exp((1 - sin(X)) .^ 2) + ...
        (X - Y) .^ 2;
end

The function can be represented in Latex as follows:

f(x, y) = sin(x)e^{(1-cos(y))^2}+cos(y)e^{(1-sin(x))^2}+(x-y)^2

References:

• Momin Jamil and Xin-She Yang, A literature survey of benchmark functions for global optimization problems, Int. Journal of Mathematical Modelling and Numerical Optimisation}, Vol. 4, No. 2, pp. 150–194 (2013), arXiv:1308.4008
• S. K. Mishra, “Global Optimization By Differential Evolution and Particle Swarm Methods: Evaluation On Some Benchmark Functions,” Munich Research Papers in Economics, Available Online: http://mpra.ub.uni-muenchen.de/1005/.