A thorough study of the performance of simulated annealing with geometric cooling in correlated and long tailed spatial scenarios

TL;DR

This paper investigates how the statistical distribution of city coordinates affects the performance of simulated annealing in solving the traveling salesman problem, revealing that distribution shape influences results more than variance.

Contribution

It introduces a method to analyze the impact of coordinate distributions on simulated annealing performance in correlated and long-tailed scenarios, highlighting the importance of distribution shape.

Findings

Performance depends on distribution shape, not variance.

Performance varies with power law decay exponents.

Improvements occur when the second moment exists, not the first.

Abstract

Metaheuristics, as the simulated annealing used in the optimization of disordered systems, goes beyond physics, and the traveling salesman is a paradigmatic NP-complete problem that allows inferring important theoretical properties of the algorithm in different random environments. Many versions of the algorithm are explored in the literature, but so far the effects of the statistical distribution of the coordinates of the cities on the performance of the algorithm have been neglected. We propose a simple way to explore this aspect by analyzing the performance of a standard version of the simulated annealing (geometric cooling) in correlated systems with a simple and useful method based on a linear combination of independent random variables. Our results suggest that performance depends on the shape of the statistical distribution of the coordinates but not necessarily on its variance corroborated by the cases of uniform and normal distributions. On the other hand, a study with different power laws (different decay exponents) for the coordinates always produces different performances. We show that the performance of the simulated annealing, even in its best version, is not improved when the distribution of the coordinates does not have the first moment. However, surprisingly, we still observe improvements in situations where the second moment is not defined but not where the first one is not defined. Finite-size scaling fits, and universal laws support all of our results. In addition, our study show when the cost must be scaled.