Mathematical Definition

Plots

Quartic Function

Quartic Function

Quartic Function

Contour of the function is presented below:

Quartic Function

Description and Features

  • The function is continuous.
  • The function is not convex.
  • The function is defined on n-dimensional space.
  • The function is multimodal.
  • The function is differentiable.
  • The function is separable.

Input Domain

The function can be defined on any input domain but it is usually evaluated on $x_i \in [-1.28, 1.28]$ for $i=1, …, n$.

Global Minima

The function has one global minimum $f(\textbf{x}^{\ast})=0 + \it\text{random noise}$ at $\textbf{x}^{\ast} = (0, …, 0)$.

Implementation

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

% Computes the value of Quartic benchmark function.
% SCORES = QUARTICFCN(X) computes the value of the Quartic function at 
% point X. QUARTICFCN 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 = quarticfcn(x)

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

The function can be represented in Latex as follows:

f(\mathbf{x})=f(x_1,...,x_n)=\sum_{i=1}^{n}ix_i^4+\text{random}[0,1)

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
  • http://www.cs.unm.edu/~neal.holts/dga/benchmarkFunction/quartic.html
  • R. Storn, K. Price, “Differntial Evolution - A Simple and Efficient Adaptive Scheme for Global Optimization over Continuous Spaces,” Technical Report no. TR-95-012, International Computer Science Institute, Berkeley, CA, 1996. [Available Online]: (R. Storn, K. Price, “Differntial Evolution - A Simple and Efficient Adaptive Scheme for Global Optimization over Continuous Spaces,” Technical Report no. TR-95-012, International Computer Science Institute, Berkeley, CA, 1996. [Available Online] : http://www1.icsi.berkeley.edu/~storn/TR-95-012.pdf