Mathematical Definition

\[f(x, y) = 10^5x^2 + y^2 -(x^2 + y^2)^2 + 10^{-5}(x^2 + y^2)^4\]

Plots

A contour of the function is 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 multimodal.
• 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 $x_i \in [-20, 20]$ for $i=1, …, n$.

Global Minima

The global minima $f(\textbf{x}^{\ast})=-24771.09375$ are located at $\mathbf{x^\ast}=(0, \pm 15)$.

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 deckkersaarts

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

MATLAB

An implementation of the Deckkers-Aarts Function with MATLAB is provided below.

% Computes the value of the Deckkers-Aarts function.
% SCORES = DECKKERSAARTSFCN(X) computes the value of the Deckkers-Aarts  
% function at point X. DECKKERSAARTSFCN 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 = deckkersaartsfcn(x)
    n = size(x, 2);
    assert(n == 2, 'The Deckkers-Aarts function is defined only on the 2-D space.')
    X = x(:, 1);
    Y = x(:, 2);
    
    scores = (100000 * X.^2) + Y.^2 + - (X.^2 + Y.^2).^2 + (10^-5) * (X.^2 + Y.^2 ) .^4;
end 

The function can be represented in Latex as follows:

f(x, y) = 10^5x^2 + y^2 -(x^2 + y^2)^2 + 10^{-5}(x^2 + y^2)^4

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
• M. M. Ali, C. Khompatraporn, Z. B. Zabinsky, “A Numerical Evaluation of Several Stochastic Algorithms on Selected Continuous Global Optimization Test Problems,” Journal of Global Optimization, vol. 31, pp. 635-672, 2005.