Simulated Annealing Particle Swarm Optimization Algorithm

Simulated Annealing Particle Swarm Optimization (SA-PSO) is a hybrid algorithm that combines two classical optimization methods. It enhances the global search capability of traditional Particle Swarm Optimization (PSO) by introducing a simulated annealing mechanism, making it particularly suitable for solving multimodal function extremum problems.

In standard PSO, each particle updates its position and velocity by tracking individual and global optimal solutions, but it tends to fall into local optima. SA-PSO introduces the probabilistic acceptance criterion of simulated annealing after each iteration: even if a new solution is inferior to the current one, it still has a certain probability of being accepted, helping the algorithm escape local optimal regions. In code implementation, this involves comparing fitness values and using a temperature-dependent probability calculation (e.g., p = exp(-Δf/T)) to decide whether to accept worse solutions.

The test functions used in the example program - Camel, Rastrigrin, and Ackley - are all classic multimodal optimization problems. For instance: The Camel function contains multiple local minima, where traditional PSO may get trapped near non-global optima; The periodic oscillatory nature of the Rastrigrin function demands stronger local optimum escaping capabilities from algorithms; The Ackley function hides its global optimum within flat regions, requiring a balance between exploration and exploitation.

SA-PSO controls the annealing process through temperature parameters: the high-temperature phase emphasizes global exploration, while gradually shifting to local refinement as temperature decreases. This dynamic adjustment allows the algorithm to quickly locate potential optimal regions while avoiding premature convergence. Implementation typically involves a cooling schedule (e.g., T = T0 * α^k) and temperature-aware velocity updates in the PSO component.