Discrete comprehensive learning particle swarm optimization algorithm with Metropolis acceptance criterion for traveling salesman problem

Particle swarm optimization (PSO) algorithm, one of the most popular swarm intelligence algorithms, has been widely studied and applied to a large number of continuous and discrete optimization problems. In this paper, a discrete comprehensive learning PSO algorithm, which uses acceptance criterion of simulated annealing algorithm, is proposed for Traveling Salesman Problem (TSP). A new flight equation, which can learn both from personal best of each particle and features of problem at hand, is designed for TSP problem. Lazy velocity, which is calculated in each dimension only when needed, is proposed to enhance the effectiveness of velocity. Eager evaluation, which evaluates each intermediate solution after velocity component is applied to the solution, is proposed to search the solution space more finely. Aiming to enhance its ability to escape from premature convergence, particle uses Metropolis acceptance criterion to decide whether to accept newly produced solutions. Systematic experiments were carried to show the advantage of the new flight equation, to verify the necessity to use non-greedy acceptance strategy for keeping sufficient diversity, and to compare lazy velocity and eager velocity. The comparison, carried on a wide range of benchmark TSP problems, has shown that the proposed algorithm is better than or competitive with many other state-of-the-art algorithms.