Mathematical Definition

\[f(x,y) = -\frac{1}{30}e^{2 \left |1 - \frac{\sqrt{x^2 + y^2}}{\pi} \right |} \cos^2(x) \cos^2(y)\]

Plots

Description and Features

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

Input Domain

The function can be defined on any input domain but it is usually evaluated on $x_i \in [-10, 10]$ for $i=1, 2$.

Global Minima

The function has a global minimum at $f(\textbf{x}^{\ast})=-24.15681551650653 $ located at $\mathbf{x^\ast}=(\pm 9.646157266348881 , \pm 9.646157266348881)$.

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 carromtable

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

MATLAB

An implementation of the Carrom Table Function with MATLAB is provided below.

function scores = carromtablefcn(x)
    n = size(x, 2);
    assert(n == 2, 'The Carrom Table function is only defined on a 2D space.')
    X = x(:, 1);
    Y = x(:, 2);

    s = 1 - ((sqrt(X.^2 + Y.^2)) / pi);
    scores = - (1/30) * exp(2 * abs(s)) .* (cos(X).^2) .* (cos(Y).^2);

The function can be represented in Latex as follows:

f(x,y) = -\frac{1}{30}e^{2 \left |1 - \frac{\sqrt{x^2 + y^2}}{\pi} \right |} \cos^2(x) \cos^2(y)

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