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We study the convergence rate of continuous-time simulated annealing $(X_t;\, t \ge 0)$ and its discretization $(x_k; \, k =0,1, \ldots)$ forapproximating the global optimum of a given function $f$. We prove that thetail probability $\mathbb{P}(f(X_t) > \min f +\delta)$ (resp.$\mathbb{P}(f(x_k) > \min f +\delta)$) decays polynomial in time (resp. incumulative step size), and provide an explicit rate as a function of the modelparameters. Our argument applies the recent development on functionalinequalities for the Gibbs measure at low temperatures -- the Eyring-Kramerslaw. In the discrete setting, we obtain a condition on the step size to ensurethe convergence.

Simulated annealing from continuum to discretization: a convergence analysis via the Eyring--Kramers law