Convergence Assessment for Ant Colony Optimization Based on Maximum Singular Values of Solution Matrix and Pheromone Matrix

Ant colony optimization (ACO) algorithm is a swarm intelligence approach widely applied to real-world optimization problems. However, how to access the convergence status of ACO remains a challenging and open issue, limiting the development of the ACO. To address this issue, this paper introduces a convergence assessment method based on matrix analysis, integrating the maximum singular values of both the solution matrix and the pheromone matrix to comprehensively quantify the algorithm’s convergence behavior, with the advantages in two-fold. First, the maximum singular value of the solution matrix measures the diversity of the population, indicating the extent of the search space explored by the ant colony. Second, the maximum singular value of the pheromone matrix evaluates the preference and stability of path selection, showing the diversity of generated solutions based on pheromone. Experimental validation on traveling salesman problem datasets shows that these two metrics complement each other in assessing the convergence dynamics of ACO, effectively capturing both solution concentration trends and pheromone distribution stability. Therefore, this research offers new insights for performance analysis and parameter optimization in matrix-based ACO algorithms.