Mathematical Definition

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

Plots

A contour of the function is presented below:

Description and Features

• The function is 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 $x_i \in [-20, 0]$ for $i=1, 2$.

Global Minima

The function has one global minimum at $f(\textbf{x}^{\ast})= e^{-200}$ located at $\mathbf{x^\ast}=(-10, -10)$.

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 brent

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

MATLAB

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

% Computes the value of the Egg Crate function.
% SCORES = BRENTFCN(X) computes the value of the Brent 
% function at point X. BRENTFCN 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 = brentfcn(x)
    n = size(x, 2);
    assert(n == 2, 'The Brent function is defined only on the 2-D space.')
    X = x(:, 1);
    Y = x(:, 2);
    
    scores = (X + 10).^2 + (Y + 10).^2 + exp(-X.^2 - Y.^2);
end

The function can be represented in Latex as follows:

f(x, y) = (x + 10)^2 + (y + 10)^2 + e^{-x^2 - 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