# Mathematical Definition

In the above definition, $\alpha$ is a real-valued parameter. For the value of $\alpha=\frac{1}{8}$, the contour of the function resembles a smiling cat, leading to the naming of the function.

# Plots

For the value of $\alpha=1/8$, the function looks as:    A contour of the function is presented below: # Description and Features

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

# Input Domain

The function can be defined on any input domain but it is usually evaluated on $x_i \in [-2, 2]$ for $i=1, …, n$.

# Global Minima

The function has one global minimum at $f(\textbf{x}^{\ast}) = 0$ located at $\mathbf{x^\ast}=(-1, …, -1)$.

# Implementation

An implementation of the Happy Cat Function with MATLAB is provided below.

The function can be represented in Latex as follows:

# Acknowledgement:

• Hans-Georg Beyer and Steffen Finck, HappyCat – A Simple Function Class Where Well-Known Direct Search Algorithms Do Fail, Parallel Problem Solving from Nature - PPSN XII, pp. 367–376 (2012), https://doi.org/10.1007/978-3-642-32937-1_37