Asymptotic Convergence of Simulated Annealing

Abstract

In Chapter 7 we presented several popular metaheuristics for finding high-quality local optima. Their popularity can be attributed mainly to the many practical successes they achieved for a wide variety of problems. An interesting question is whether the good performance of the metaheuristics can also be supported by theoretical results. This is the case for simulated annealing, for which it has been shown that it asymptotically converges to the set of globally optimal solutions if it is applied on a neighborhood graph that is finite, symmetric, and strongly connected. This chapter is devoted to proving this result. In Section 8.6 we show that the result does not necessarily hold for tabu search. To put the convergence result of simulated annealing in the right perspective, we stress that it only states that the algorithm reaches a global optimum with probability 1 after an infinite number of iterations. This means that the simple total enumeration algorithm, which takes exponential but finite time, is more efficient when optimality has to be guaranteed. Furthermore, if the naive random search algorithm discussed in Section 7.1 stores its best solution encountered, then it also converges to a globally optimal solution. The relevance of the convergence result for simulated anneal-ing is more subtle, however. It states that, when moving through a neighborhood graph, simulated annealing searches its way to the set of globally optimal solutions. This supports the assumption that the applied search strategy is effective.