PARTIAL SIMULATED ANNEALING

Let (L(theta))(theta is an element of N) be a family of elliptic diffu sion operators on a compact and connected smooth manifold M, whose ter ms of first order are indexed by a parameter theta living in N, the n- dimensional torus. For each fixed theta, we associate to L(theta) its invariant probability mu(theta). Let f be a smooth function on M x N a nd define for theta is an element of N, F(theta) = integral f(x,theta) mu(theta)(dx). We study partial simulated annealing algorithms (using only quite directly L(theta) and f) to find the global minima of F. Th is paper presents a new proof of the convergence of these algorithms, using n + 2 partial entropies associated naturally to the problem. Thi s approach is simpler than the one exposed previously in (Miclo, 1994) , which furthermore was restricted to the case n = 1, but we need to s peed up much more the diffusion interacting with the simulated anneali ng algorithm (and in practice, this is embarrassing).