Schwefel 2.21

Mathematical Definition

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

Latex

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

Description and Features

Dimensions: d

The function has one global minimum. It is continuous, convex and unimodal. The plot shows its two-dimensional form.

• 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 $i = 1, …, d$ .

Global Minima

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

Implementation

Python Code

def function(x):
    x = np.array(x)
    return np.max(np.abs(x))

            

Solutions

References:

• http://benchmarkfcns.xyz/benchmarkfcns/schwefel221fcn.html
• 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.