Bukin no.6

Description [view source]

The Bukin function No. 6 function is part of a group of functions (see Sources) and is a well-known benchmark for testing optimization algorithms. It is mathematically defined as:

$$ \LARGE f(x, y) = 100 \sqrt{|y - 0.01x^2|} + 0.01 |x + 10| $$

The function is defined in a two-dimensional space, and is typically evaluated over $x$ $\epsilon$ $[-15, 5]$, $y$ $\epsilon$ $[-3, 3]$.

Visualization

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

Bukin6().visualize()

Features

1. Non-convexity: The function has alternating high and low regions near the global minimum inside a narrow valley. Finding the valley is trivial, but converging to the global minimum is challenging.
2. Global minimum: The global minimum is $f(x^{*}=<-10, 1>)=0$.
3. Discontinuous derivative: The function has a discontinuous derivative (e.g., the absolute values cannot be differentiated at their peaks), which poses a challenge for gradient-based optimization methods.

Discussion

This function poses a similar challenge as Rosenbrock (click here) with it's narrow valley which is easy to find, but difficult to navigate. While Rosenbrock's valley is challenging because of it's near-zero gradient, Bukin no.6's is challenging because of it's seesaw-like ups and downs, which can easily trap greedy algorithms.

Finally, the position of it's global minimum (i.e., $<-10,1>$) naturally diminishes the impact of the origin bias that some PSO-inspired algorithms tend to show.

Sources

Historical information about the Bukin family of functions does not appear to be readily available in the Enlish-speaking literature. However, it can be traced back to a 1997 article by a Russian scientist named A. D. Bukin titled New Minimization Strategy for Non-Smooth Functions : Bukin (1997).pdf