Styblinski‐Tang

Description

The Styblinski-Tang function is a well-known benchmark for testing optimization algorithms due to its complex landscape with multiple local minima. It is defined mathematically as:

$$ \LARGE f(\vec{x}) = \frac{1}{2} \sum_{i=1}^{d} \left(x_i^4 - 16x_i^2 + 5x_i\right) $$

with $d$ representing the dimensionality of the input vector $\vec{x}$. The default dimensionality (StyblinskiTang.dimensionality) is 2.

Visualization

Here is a link to an interactive Desmos version: https://www.desmos.com/3d/eb6rnxnwmk

StyblinskiTang().visualize()

Features

1. Multimodality: The function has multiple local minima, creating a challenging optimization landscape.
2. Global minimum: The global minimum for the function is $f(x^* = <-2.903534, ..., -2.903534> = -39.16617$.
3. Search space: Typically, the function is evaluated over the bounded region $[-5, 5]$ for each dimension, which is also the default value.

Discussion

The Styblinski-Tang function is useful for evaluating the performance of optimization algorithms in terms of their ability to navigate and converge in a complex, multimodal landscape. Its multiple local minima provide a rigorous test for optimization techniques, particularly those that rely on global search capabilities. Algorithms that can effectively handle the Styblinski-Tang function's landscape are likely to perform well on a variety of real-world optimization problems.

Sources

The function was introduced in the following article :

Styblinski, M. A., & Tang, T. S. (1990). Experiments in nonconvex optimization: stochastic approximation with function smoothing and simulated annealing. Neural Networks, 3(4), 467-483.

It is available here : https://www.sciencedirect.com/science/article/pii/089360809090029K