α Annealing of ant colony optimization in the infinite-range Ising model

Ant colony optimization leverages the parameter α to modulate the decision function's sensitivity to pheromone levels, balancing the exploration of diverse solutions with the exploitation of promising areas. Identifying the optimal value for α and establishing an effective annealing schedule remain significant challenges, particularly in complex optimization scenarios. This study investigates the α-annealing process of the linear ant system within the infinite-range Ising model to address these challenges. Here, "linear" refers to the decision function employed by the ants. By systematically increasing α, we explore its impact on enhancing the search for the ground state. We derive the Fokker-Planck equation for the pheromone ratios and obtain the joint probability density function (PDF) in stationary states. As α increases, the joint PDF transitions from a monomodal to a multimodal state. In the homogeneous fully connected Ising model, α-annealing facilitates the transition from a trivial solution at α=0 to the ground state. The parameter α in the annealing process plays a role analogous to the transverse field in quantum annealing. Our findings demonstrate the potential of α-annealing in navigating complex optimization problems, suggesting its broader application beyond the infinite-range Ising model.