Mathematical Definition

\[f(x, y)=0.5 + \frac{sin^2(cos(|x^2-y^2|))-0.5}{(1+0.001(x^2+y^2))^2}\]

Plots

Two contours of the function are as presented below:

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 not separable.

Input Domain

The function can be defined on any input domain but it is usually evaluated on $x_i \in [-100, 100]$ for $i=1, 2$.

Global Minima

The function has one global minimum $f(\textbf{x}^{\ast})=0.00156685$ at $\textbf{x}^{\ast} = (0, 1.253115)$.

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 schaffer3

print(schaffer3([[0, 0],
              [1, 1]]))

MATLAB

An implementation of the Schaffer N. 3 Function with MATLAB is provided below.

% Computes the value of the Schaffer N. 3 function.
% SCORES = SCHAFFERN3FCN(X) computes the value of the Schaffer N. 3  
% function at point X. SCHAFFERN3FCN 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 = schaffern3fcn(x)
    n = size(x, 2);
    assert(n == 2, 'Schaffer function N. 3 is only defined on a 2D space.')
    X = x(:, 1);
    Y = x(:, 2);
    
    numeratorcomp = (sin(cos(abs(X .^ 2 - Y .^ 2))) .^ 2) - 0.5; 
    denominatorcomp = (1 + 0.001 * (X .^2 + Y .^2)) .^2 ;
    scores = 0.5 + numeratorcomp ./ denominatorcomp;
end

The function can be represented in Latex as follows:

f(x, y)=0.5 + \frac{sin^2(cos(|x^2-y^2|))-0.5}{(1+0.001(x^2+y^2))^2}

References:

• S. K. Mishra, “Some New Test Functions For Global Optimization And Performance of Repulsive Particle Swarm Method,” [Available Online]: http://mpra.ub.uni-muenchen.de/2718/
• 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