We investigate how to tune a generalized simulated annealing
algorithm with piecewise constant cooling schedule to get an optical
convergence exponent. The optimal convergence exponent of generalized
simulated annealing algorithms has been computed by Catoni and Trouvé. It is reached only with triangular sequences of temperatures, meaning that different
finite sequences are used, depending on the time resource available for
computations (expressed by an overall number of iterations). We show first that
it is possible to get close to the optimal convergence exponent uniformly over
suitably bounded families of energy landscapes using a fixed number of
temperature steps. Then we show that, letting the number of steps increase with
the time resource, we can build a cooling schedule which is universally robust
with respect to the convergence exponent: a fixed triangular sequence of
temperatures gives an optimal convergence exponent for any energy landscape.
Piecewise constant temperature sequences are often used in practice: in
favourable cases, the use of the same temperature during a large number of
iterations allows tabulating the exponential penalties appearing in the
transition matrix, thus sparing a significant amount of computer time. The
proofs we give rely on Freidlin and Wentzell's closed formulas for the exit
time and point from subdomains of time homogeneous Markov chains.