Mathematical Definition

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

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 $x \in [-1, 2]$ and $y \in [-1, 1]$.

Global Minima

On the on $x \in [-1, 2]$ and $x \in [-1, 1]$ cube, the global minimum $f(\textbf{x}^{\ast})=-2.02181$ is located at $\mathbf{x^\ast}=(0, 0)$.

Implementation

Python

For Python, the function is implemented in the benchmarkfcns package and can be installed from command line with pip install benchmarkfcns.

from benchmarkfcns import adjiman

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

MATLAB

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

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

The function can be represented in Latex as follows:

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

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
• C. S. Adjiman, S. Sallwig, C. A. Flouda, A. Neumaier, “A Global Optimization Method, aBB for General Twice-Differentiable NLPs-1, Theoretical Advances,” Computers Chemical Engineering, vol. 22, no. 9, pp. 1137-1158, 1998.
• Qing, A., “Differential Evolution: Fundamentals and Applications in Electrical Engineering”, Wiley, 2009. https://books.google.com/books?id=Pp-SHz6dIJ0C