Mathematical Definition

Plots

Schwefel 2.21 Function

Schwefel 2.21 Function

Schwefel 2.21 Function

Contour of the function is presented below:

Schwefel 2.21 Function

Description and Features

  • The function is continuous.
  • The function is convex.
  • The function is defined on n-dimensional space.
  • The function is unimodal.
  • The function is non-differentiable.
  • The function is separable.

Input Domain

The function can be defined on any input domain but it is usually evaluated on $x_i \in [-100, 100]$ for $i=1, …, n$.

Global Minima

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

Implementation

An implementation of the Schwefel 2.21 Function with MATLAB is provided below.

% Computes the value of the Schwefel 2.21 function.
% SCORES = SCHWEFEL221FCN(X) computes the value of the Schwefel 2.21 
% function at point X. SCHWEFEL221FCN 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.
% 
% 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 = schwefel221fcn(x)
    scores = max(abs(x), [], 2);
end

The function can be represented in Latex as follows:

f(\mathbf{x})=f(x_1, ..., x_n)=\max_{i=1,...,n}|x_i| 

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
  • H. P. Schwefel, “Numerical Optimization for Computer Models,” John Wiley Sons, 1981.