Mathematical Definition

In this formula, the parameters $a$ and $b$ are constants and are generally set to $a=1$ and $b=100$.

Plots

Rosenbrock Function

Rosenbrock Function

Rosenbrock Function

The contour of the function is as presented below:

Rosenbrock Function

Description and Features

  • The function is continuous.
  • The function is convex.
  • The function is defined on n-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 $x_i \in [-5, 10]$ for $i=1, …, n$ .

Global Minima

The function has one global minimum $f(\textbf{x}^{\ast})=0$ at $\textbf{x}^{\ast} = (1, …, 1)$.

Implementation

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

% Computes the value of the Rosenbrock benchmark function.
% SCORES = ROSENBROCKFCN(X) computes the value of the Rosenbrock function  
% at point X. ROSENBROCKFCN accepts a matrix of size M-by-N 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: 
% https://en.wikipedia.org/wiki/Rosenbrock_function
% 
% 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 = rosenbrockfcn(x)
    scores = 0;
    n = size(x, 2);
    assert(n >= 1, 'Given input X cannot be empty');
    a = 1;
    b = 100;
    for i = 1 : (n-1)
        scores = scores + (b * ((x(:, i+1) - (x(:, i).^2)) .^ 2)) + ((a - x(:, i)) .^ 2);
    end
end

The function can be represented in Latex as follows:

f(x, y)=\sum_{i=1}^{n}[b (x_{i+1} - x_i^2)^ 2 + (a - x_i)^2]

References: