Mean square rates of convergence in the continuous time simulated annealing algorithm on \({\mathbb{R}}^ d\) (Q1099503)

To locate a global minimum value of a function V at \(\theta\) one may employ the simulated annealing algorithm, a gradient descent procedure modified to climb uphill in order to escape local minima. The continuous version of the simulated annealing procedure is given by the diffusion \[ dX_ t=-\nabla V(X_ t) dt+\sigma_ t dW_ t. \] The paper shows under suitable assumptions on V if \[ \sigma^ 2_ t=c/\log (t+2), \] then there exist positive constants \(c_ 1\) and \(c_ 2\) such that \[ c_ 2\leq \sqrt{\log t} E| X_ t-\theta |^ 2\leq c_ 1 \] for sufficiently large t.