Simulated annealing algorithm is a classical nature-inspired computational method of imitating physics, which has achieved tremendous results in algorithm design and application researches. Simulated annealing strategy is also widely integrated into researches of modern swarm intelligence evolutionary algorithms. Early theoretical researches for the algorithm on performance analysis and convergence analysis were mainly based on the Markov chain theory in the random process, and obtained the convergence theorem in probability. However, it is difficult to analyze the actual searching behaviors of the algorithm using the Markov chain theory. Also, it's hard to get some improvement strategies that are useful for the algorithm. The main attempt is to analyze the operating mechanism and the convergence of the simulated annealing algorithm by using dynamical system theory in this paper. Due to mathematics and physics have a solid theoretical foundations and a variety of analytical methods, they can be used to analyzing and modeling this class of random heuristic algorithms theoretically, so that directing designation and improvement of algorithms based on problem characteristics. Therefore, by comparing the searching process of the algorithm to the elastic motion of a particle, the change of the function value during the running of the simulated annealing algorithm can be regarded as the particle doing simple harmonic vibration or damping vibration. So, the dynamical system models of the ordinary differential equation are established for the algorithm according to the principle of elastic mechanics and the algorithm running mechanism. And then, the models are solved and analyzed by using the stability theory of ordinary differential equations. The local convergence of the simulated annealing algorithm in the early-and-mid stage and the global convergence in the late stage are proved, and a rational theoretical explanation of its operating mechanism is stated. Hereafter, based on the dynamical system models, the relationship between the damping coefficient of the model equation and the convergence speed of the algorithm is derived. Furthermore, the convergence speed of the algorithm is inferred, which is associated with the annealing temperature. Based on the theoretical analyses, in response to the drawbacks of the tempering mechanism which usually relied on the experience before, a simple and easy tempering time criterion is proposed in order to improve global searching performance of the simulated annealing algorithm. That is, when the elastic coefficient tends to a small value, it can be used as a tempering point for tempering strategy. Experiments applying the primitive simulated annealing algorithm are implemented in order to verify the theoretical results on several typical test problems. Firstly, these experiments show that the numerical convergence lines coincide with the convergence results of theoretical analysis. Secondly, experiments verify the changing tendency of the convergence speed along with annealing temperature in theory is consistent with the actual numerical experiment. As the annealing temperature decreases, the damping coefficient increases and the attenuation factor decreases, and then the convergence speed increases. Thirdly, the experiments also show the validity of the proposed tempering moment criterion on the benchmark problem. In the end, theoretical and experimental analysis demonstrates that the dynamical system models established in this paper are suitable for describing the simulated annealing algorithm. © 2019, Science Press. All right reserved.
