On the complexity to approach optimum solutions by inhomogeneous Markov chains

We analyse the probability 1 - delta to be in an optimum solution after k steps of an inhomogeneous Markov chain which is specified by a logarithmic cooling schedule c(k) = Gamma/ln (k + 2). We prove that after k > (n/delta)(O(Gamma)) steps the probability to be in an optimum solution is larger than 1 - delta, where n is an upper bound for the size of local neighbourhoods and Gamma is a parameter of the entire configuration space. By counting the occurrences of configurations, we demonstrate for an application with known optimum solutions that the lower bound indeed ensures the stated probability for a relatively small constant in O(Gamma).