# Periodic Function

# Mathematical Definition

# Plots

Two contours of the function are presented below:

# 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 non-separable.

# Input Domain

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

# Global Minima

The function has on global minimum $f(\mathbf{x}^{\ast})=0.9$ at $\mathbf{x}^{\ast}=(0 … 0)$.

# Implementation

An implementation of the **Periodic Function** with `MATLAB`

is provided below. The function can be implemented with a `for`

loop
that iterates over the input components but `MATLAB`

and `Octave`

have built-in facilities that allow a more brief implementation.

The function can be represented in Latex as follows:

# 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
- M. M. Ali C. Khompatraporn Z. B. Zabinsky “A Numerical Evaluation of Several Stochastic Algorithms on Selected Continuous Global Optimization Test Problems,” Journal of Global Optimization vol. 31 pp. 635-672 2005.