Mathematical Definition

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Brent Function

Brent Function

Brent Function

A contour of the function is presented below: Brent Function

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

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.
% For more information, please visit:
% benchmarkfcns.xyz/fcns/brentfcn
% 
% 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