necessary and sufficient conditions for success of the metropolis algorithm for optimization
This paper focusses on the performance of the Metropolis algorithm when employed for solving combinatorial optimization problems. One finds in the literature two notions of success for the Metropolis algorithm in the context of such problems. First, we show that both these notions are equivalent. Next, we provide two characterizations, or in other words, necessary and sufficient conditions, for the success of the algorithm, both characterizations being conditions on the family of Markov chains which the Metropolis algorithm gives rise to when applied to an optimization problem. The first characterization is that the Metropolis algorithm is successful if in every chain, for every set A of states not containing the optimum, the ratio of the ergodic flow out of A to the capacity of A is high. The second characterization is that in every chain the stationary probability of the optimum is high and that the family of chains mixes rapidly. We illustrate the applicability of our results by giving alternative proofs of certain known results.
metropolis algorithm optimization success of markov chain families rapid mixing of markov chains