$\alpha$ Annealing of Ant Colony Optimization in the infinite-range Ising model

Ant colony optimization (ACO) leverages the parameter $\alpha$ 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 $\alpha$ and establishing an effective annealing schedule remain significant challenges, particularly in complex optimization scenarios. This study investigates the $\alpha$-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 $\alpha$, 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 $\alpha$ increases, the joint PDF transitions from a mono-modal to a multi-modal state. In the homogeneous fully connected Ising model, $\alpha$-annealing facilitates the transition from a trivial solution at $\alpha=0$ to the ground state. The parameter $\alpha$ in the annealing process plays a role analogous to the transverse field in quantum annealing. Our findings demonstrate the potential of $\alpha$-annealing in navigating complex optimization problems, suggesting its broader application beyond the infinite-range Ising model.