# Quartic Function

# Mathematical Definition

# Plots

Contour of the function is 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 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.

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
- 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