Constructing efficient simulated annealing algorithms

Simulated annealing is a general purpose optimization technique for combinatorial optimization problems. In order to improve the efficiency of simulated annealing, the neighbourhood structure and the generation probabilities have to be designed carefully. Usually, this process is problem dependent and, hence, unlikely to be generally applicable to various combinatorial problems. In the present paper a general theory is introduced and proved to support the construction of efficient generation mechanisms. In order to reduce the complexity of the problem, a particular problem-dependent equivalence relation within the set S of all possible configurations can be defined, determining a quotient set within the original set S. If all optimal configurations are assigned to the same equivalence class, in order to find an optimal configuration, it suffices to wander within the quotient set instead of the original set S. If the average size of a non-optimal equivalence class is much larger than the size of the optimal class, the optimal configurations will be preferred during the annealing process. This translates to a significant reduction in CPU time. To avoid the need to compute the quotient set explicitly, the search procedure in the quotient set is simulated within the original set of configurations. Our approach has parallels with the ejection chain strategies used in tabu search, and our theory provides further motivation for such strategies (in addition to the combinatorial leverage theorems of tabu search). Applications to the channel assignment problem which occurs during the design of cellular radio systems demonstrate clearly the power of the general construction theory. With the new method we can construct generation mechanisms that reduce the exponential complexity of the configuration set to a polynomial dependence on the problem size, as characteristic of the combinatorial leverage phenomenon of ejection chains.