Simulated annealing algorithms and Markov chains with rare transitions

In these notes, written for a D.E.A. course at University Paris XI during the first term of 1995, we prove the essentials about stochastic optimisation algorithms based on Markov chains with rare transitions, under the weak assumption that the transition matrix obeys a large deviation principle. We present a new simplified line of proofs based on the Freidlin and Wentzell graphical approach. The case of Markov chains with a periodic behaviour at null temperature is considered. We have also included some pages about the spectral gap approach where we follow Diaconis and Stroock [13] and Ingrassia [23] in a more conventional way, except for the application to non reversible Metropolis algorithms (subsection 6.2.2) where we present an original result.