Mathematical Definition

\[f(\textbf{x})=f(x_1, ..., x_n)=\sum_{i=1}^n x_i^{2}+(\sum_{i=1}^n 0.5ix_i)^2 + (\sum_{i=1}^n 0.5ix_i)^4\]

Plots

The contour of the function:

Description and Features

• The function is continuous.
• The function is convex.
• The function can be defined on n-dimensional space.
• The function is unimodal.

Input Domain

The function can be defined on any input domain but it is usually evaluated on the hypercube $x_i \in [-5, 10]$ for $i = 1…n$.

Global Minima

$f(\textbf{x}^{\ast}) = 0$ at $\textbf{x}^{\ast} = (0, …, 0)$

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 zakharov

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

MATLAB

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

% Computes the value of Zakharov benchmark function.
% SCORES = ZAKHAROVFCN(X) computes the value of the Zakharov function at 
% point X. ZAKHAROVFCN 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
% each 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 = zakharovfcn(x)

    n = size(x, 2);
    comp1 = 0;
    comp2 = 0;
    
    for i = 1:n
        comp1 = comp1 + (x(:, i) .^ 2);
        comp2 = comp2 + (0.5 * i * x(:, i));
    end
     
    scores = comp1 + (comp2 .^ 2) + (comp2 .^ 4);
end

The function can be represented in Latex as follows:

f(\textbf{x})=f(x_1, ..., x_n)=\sum_{i=1}^n x_i^{2}+(\sum_{i=1}^n 0.5ix_i)^2 + (\sum_{i=1}^n 0.5ix_i)^4

References:

• http://www.sfu.ca/~ssurjano/zakharov.html