# Mathematical Definition

In this formula, $d$ and $\alpha$ are constants and are usually set to $d = 1, \alpha=0.5$.

# Plots

For $d=2, \alpha=0.1$, the plots are:

For $d=2, \alpha=2$, the plots are:

Two contours of the function are presented below: For $d=2, \alpha=0.1$, the function contour is:

For $d=2, \alpha=2$, the function contour is:

# 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. It is evaluated on $x_i \in [-5, 5]$ for $i=1, 2$.

# Global Minima

The global minimum of the function depends on the hypercube it is defined on. On the hypercube $[-\gamma, \gamma]^n$, $f(\textbf{x}^{\ast})= -\gamma$ located at $\mathbf{x^\ast}=(-\gamma, 0, …, 0)$.

# Implementation

An implementation of the Ridge Function with MATLAB is provided below.

The function can be represented in Latex as follows:

# References:

• Beyer HG., Finck S. (2012) HappyCat – A Simple Function Class Where Well-Known Direct Search Algorithms Do Fail. In: Coello C.A.C., Cutello V., Deb K., Forrest S., Nicosia G., Pavone M. (eds) Parallel Problem Solving from Nature - PPSN XII. PPSN 2012. Lecture Notes in Computer Science, vol 7491. Springer, Berlin, Heidelberg, https://doi.org/10.1007/978-3-642-32937-1_37
• Oyman, A.I.: Convergence Behavior of Evolution Strategies on Ridge Functions. Ph.D. Thesis, University of Dortmund, Department of Computer Science (1999)