Mathematical Definition

\[f(x, y) = (x^{2} + y - 11)^{2} + (x + y^{2} - 7)^{2}\]

Plots

The contour of the function:

Description and Features

• The function is continuous.
• The function is not convex.
• The function is defined on the 2-dimensional space.
• The function is multimodal.

Input Domain

The function can be defined on any input domain but it is usually evaluated on $x_i \in [-6, 6]$ for $i = 1, 2$.

Global Minima

The function has four local minima at:

• $f(\textbf{x}^{\ast}) = 0$ at $\textbf{x}^{\ast} = (3, 2)$
• $f(\textbf{x}^{\ast}) = 0$ at $\textbf{x}^{\ast} = (-2.805118, 3.283186)$
• $f(\textbf{x}^{\ast}) = 0$ at $\textbf{x}^{\ast} = (-3.779310, -3.283186)$
• $f(\textbf{x}^{\ast}) = 0$ at $\textbf{x}^{\ast} = (3.584458, -1.848126)$

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 himmelblau

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

MATLAB

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

% Computes the value of the Himmelblau's benchmark function.
% SCORES = HIMMELBLAUFCN(X) computes the value of the Himmelblau's
% function at point X. HIMMELBLAUFCN 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: 
% https://en.wikipedia.org/wiki/Himmelblau's_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 = himmelblaufcn(x)
    n = size(x, 2);
    assert(n == 2, 'Himmelblau''s function is only defined on a 2D space.')
    X = x(:, 1);
    Y = x(:, 2);
    
    scores = ((X .^ 2 + Y - 11) .^2) + ((X + (Y .^ 2) - 7) .^ 2);
end

The function can be represented in Latex as follows:

f(x, y) = (x^{2} + y - 11)^{2} + (x + y^{2} - 7)^{2}

References:

• https://en.wikipedia.org/wiki/Himmelblau%27s_function