Annealing diffusions in a slowly growing potential

Pierre-André Zitt (1)

1 MODAL'X - Modélisation aléatoire de Paris X

We consider a continuous analogue of the simulated annealing algorithm in $R^d$. We prove a convergence result, under hypotheses weaker than the usual ones. In particular, we cover cases where the gradient of the potential goes to zero at infinity. The proof follows an idea of L. Miclo, but we replace the Poincaré and log-Sobolev inequalities (which do not hold in our setting) by weak Poincaré inequalities. We estimate the latter with measure-capacity criteria. We show that, despite the absence of a spectral gap, the convergence still holds for the "classical" schedule t = c/ ln(t), if c is bigger than a constant related to the potential.

Mots clés: simulated annealing; weak Poincaré inequality

Domaines: Probabilités [math.PR]

Pierre-André Zitt. Annealing diffusions in a slowly growing potential. Stochastic Processes and their Applications, 2008, 118 (1), pp.76-119. ⟨10.1016/j.spa.2007.04.002⟩. ⟨hal-00084081⟩