Verification and rectification of the physical analogy of simulated annealing for the solution of the traveling salesman problem

M. Hasegawa

Phys. Rev. E 83, 036708 – Published 31 March 2011

Abstract

The aim of the present study is to elucidate how simulated annealing (SA) works in its finite-time implementation by starting from the verification of its conventional optimization scenario based on equilibrium statistical mechanics. Two and one supplementary experiments, the design of which is inspired by concepts and methods developed for studies on liquid and glass, are performed on two types of random traveling salesman problems. In the first experiment, a newly parameterized temperature schedule is introduced to simulate a quasistatic process along the scenario and a parametric study is conducted to investigate the optimization characteristics of this adaptive cooling. In the second experiment, the search trajectory of the Metropolis algorithm (constant-temperature SA) is analyzed in the landscape paradigm in the hope of drawing a precise physical analogy by comparison with the corresponding dynamics of glass-forming molecular systems. These two experiments indicate that the effectiveness of finite-time SA comes not from equilibrium sampling at low temperature but from downward interbasin dynamics occurring before equilibrium. These dynamics work most effectively at an intermediate temperature varying with the total search time and thus this effective temperature is identified using the Deborah number. To test directly the role of these relaxation dynamics in the process of cooling, a supplementary experiment is performed using another parameterized temperature schedule with a piecewise variable cooling rate and the effect of this biased cooling is examined systematically. The results show that the optimization performance is not only dependent on but also sensitive to cooling in the vicinity of the above effec-tive temperature and that this feature is interpreted as a consequence of the presence or absence of the workable interbasin dynamics. It is confirmed for the present instances that the effectiveness of finite-time SA derives from the glassy relaxation dynamics occurring in the “landscape-influenced” temperature regime and that its naive optimization scenario should be rectified by considering the analogy with vitrification phenomena. A comprehensive guideline for the design of finite-time SA and SA-related algorithms is discussed on the basis of this rectified analogy.