Drop‐Wave

Description [view source]

The Drop-Wave function is a commonly used benchmark for testing optimization algorithms. It is mathematically defined as:

$$ \LARGE f(x, y) = -\frac{1 + \cos(12 \sqrt{x^2 + y^2})}{0.5(x^2 + y^2) + 2} $$

The function is defined in a two-dimensional space, and is commonly evaluated over $x$ $\epsilon$ $[-5.12, 5.12]$, $y$ $\epsilon$ $[-5.12, 5.12]$.

Visualization :

Here is a link to an interactive Desmos version: https://www.desmos.com/3d/djfs4gxf3i

Dropwave().visualize()

Dropwave().visualize(plot_bounds=[(-1, 1), (-1, 1)])

Features

1. Multimodality: The cosine component introduces oscillations, which create numerous local minima and maxima, making it challenging for optimization algorithms to find the global minimum.
2. Global minimum: The global minimum is $f(x^{*}=<0,0>)=-1$.

Discussion

The Drop-Wave function is particularly useful for testing the robustness of optimization algorithms, especially those that need to navigate complex landscapes with multiple local minima. Its oscillatory nature and multiple local minima make it a difficult problem for many optimizers. This function is used to evaluate how well algorithms can handle exploration and exploitation in a highly multimodal environment.

Sources

The oldest reference we could find is (Molda & Smutnicki, 2005), but older ones may exist since the former is a review.