Particle swarm optimization (PSO) as a swarm intelligence-based optimization algorithm has been widely applied to solve various real-world optimization problems. However, traditional PSO algorithms encounter issues such as premature convergence and an imbalance between global exploration and local exploitation capabilities when dealing with complex optimization tasks. To address these shortcomings, an enhanced PSO algorithm incorporating velocity pausing and adaptive strategies is proposed. By leveraging the search characteristics of velocity pausing and the terminal replacement mechanism, the problem of premature convergence inherent in standard PSO algorithms is mitigated. The algorithm further refines and controls the search space of the particle swarm through time-varying inertia coefficients, symmetric cooperative swarms concepts, and adaptive strategies, balancing global search and local exploitation. The performance of VASPSO was validated on 29 standard functions from Cec2017, comparing it against five PSO variants and seven swarm intelligence algorithms. Experimental results demonstrate that VASPSO exhibits considerable competitiveness when compared with 12 algorithms. The relevant code can be found on our project homepage.

The ant colony algorithm faces dimensional catastrophe problems when solving the large-scale traveling salesman problem, which leads to unsatisfactory solution quality and convergence speed. To solve this problem, an adaptive ant colony optimization for large-scale traveling salesman problem (AACO-LST) is proposed. First, AACO-LST improves the state transfer rule to make it adaptively adjust with the population evolution, thus accelerating its convergence speed; then, the 2-opt operator is used to locally optimize the part of better ant paths to further optimize the solution quality of the proposed algorithm. Finally, the constructed adaptive pheromone update rules can significantly improve the search efficiency and prevent the algorithm from falling into local optimal solutions or premature stagnation. The simulation based on 45 traveling salesman problem instances shows that AACO-LST improves the solution quality by 79% compared to the ant colony system (ACS), and in comparison with other algorithms, the PE of AACO-LST is not more than 1% and the Err is not more than 2%, which indicates that AACO-LST can find high-quality solutions with high stability. Finally, the convergence speed of the proposed algorithm was tested. The data shows that the average convergence speed of AACO-LST is more than twice that of the comparison algorithm. The relevant code can be found on our project homepage.

Traditional ant colony optimization (ACO) often suffers from insufficient initial pheromone information, premature convergence, and limited adaptability across problem scales when solving the traveling salesman problem (TSP). This study proposes a multi-strategy adaptive ant colony optimization algorithm, termed MSA-ACO. Unlike hybrid ACO methods whose added strategies are governed by separate or predetermined rules, MSA-ACO uses smoothed normalized pheromone entropy as a shared internal search-state signal. This signal coordinates search-stage switching, exploitation preference, pheromone evaporation intensity, heuristic-information dependence, and the adaptive 3-opt budget under scale-dependent control profiles. A rank-weighted multi-elite initialization method and candidate-restricted hierarchical 2-opt/3-opt search are further incorporated to improve early guidance and solution refinement. Experiments on 35 TSPLIB instances show that MSA-ACO attains the best-known solution on 22 instances and achieves an average best-solution deviation of 0.17%, compared with 3.87% for ACS and 5.27% for MMAS. Its maximum deviation is 1.22%, and the deviation remains within 1% on 33 instances. Nonparametric tests confirm significantly better overall solution quality than ACS and MMAS. Under equal distance-query budgets, MSA-ACO obtains the lowest mean tour length on st70, kroC100, and kroA150, with significant improvements over ACS-based local search on kroC100 and kroA150. Component-wise ablation and sequential timing results reveal instance-dependent component effects and identify 3-opt as a major source of computational cost as well as an important refinement mechanism on the larger instance. Moreover, compared with the fixed 1000-iteration setting, the scale-segmented iteration budget reduces average running time by 28.06–83.75% across four representative instances, while still attaining the BKS on eil76 and ts225 and yielding mean deviations of approximately 0.30% and 0.20% on ch150 and kroA200, respectively. Overall, MSA-ACO provides an effective scale-aware optimization framework for symmetric TSPs, combining consistently high solution quality with controlled computational effort across different instance scales.