On geometric convergence rate of Markov search towards the fat target

Let f:A→R be a continuous function with the minimal value f⋆, where A is the compact metric space. Let {Xt}t∈N be a Markov chain which represents the global optimization process on A. We present sufficient conditions for very strong, geometric convergence mode of the form Ef(Xt)−f∗≤ct⋅(Ef(X0)−f∗), where c∈(0,1) is some constant. This convergence mode is natural if the set of global minima is fat.