Convergence and Empirical Performance of Tanh-Based Adaptive Particle Swarm Optimization

Particle Swarm Optimization (PSO) is a widely used population-based optimization method but faces challenges in premature convergence, leading to suboptimal solutions. To address this issue, this study proposes a Tanh-Based Acceleration Coefficient PSO (TB-PSO), where the acceleration coefficients are modified using the hyperbolic tangent (tanh) function. The smooth and continuous behavior of tanh enables gradual coefficient updates, limits excessive particle velocities, and maintains swarm diversity, thereby improving convergence stability and balancing exploration and exploitation. The convergence theorem analysis confirms that TB-PSO meets stability criteria before being evaluated on unimodal and multimodal benchmark functions in 10 and 30 dimensions. Its performance is compared against several PSO variants, including TVAC-PSO, SCAC-PSO, NDAC-PSO, and SAC-PSO. In the 10-dimensional experiments, TB-PSO achieves the best overall final ranking based on the average and standard deviation of best solution, ranking first for functions f₃ and f₅, second for f₂ with only a marginal difference from the best-performing method, and remaining competitive for f₁ and f₄. These results indicate superior solution quality and stable convergence. For the 30-dimensional benchmark functions, TB-PSO ranks first for f₂, second for f₅, and third for f₁, f₃, and f₄ based on the same evaluation criteria. Although its ranking decreases compared to the 10-dimensional case, TB-PSO remains competitive, reflecting the increased complexity of high-dimensional optimization problems. Overall, the results demonstrate that the tanh-based acceleration coefficient modification effectively enhances PSO performance, particularly in lower-dimensional search spaces, while maintaining robustness in higher-dimensional scenarios.