Brownian Bridge - Continuous Ant Colony Optimization (BB-CACO)

We exploit stochastic differential equations (SDEs) to derive a novel continuous ant colony optimization (CACO) path exploration scheme that is, by definition, designed to leave the nest and return to it. The usage of Brownian bridges, hence our name Brownian bridge ACO (BB-CACO), forces the ant agent to revert back to the nest in a more 'linear' manner, resulting in lower path variance and faster return to nest. The key insight is that by increasing the diffusion parameter(s) in the Itô bridges, the ant agents can cover a much wider search space at no additional computational burden. Unlike traditional ACO, BB-ACO can explore wider search spaces without increasing the number of ant agents, hence not requiring exponentially increasing computational resources. Finally, an additional advantage is that certain classes of these SDEs admit analytical solutions under specific conditions (e.g., linear drift coefficients with constant diffusion terms), as demonstrated in [13] for quadratic loss functions and [21] for convex optimization landscapes. Finally, the other benefit is that such SDEs can also be solved for, analytically in certain circumstances, and so provide deeper insights into the geometry of the global minimum and the resulting optimal solution, if they exist, or alternatively the optimal approximation. Coupled with the metaheuristic overlay of the ACO process, BB-CACO is shown to be a powerful addition to the existing continuous ACO algorithms and opens up new approaches to novel research.