Diffusion for Global Optimization in $\mathbb{R}^n $

Global optimization of robust chance constrained problems

“…[20] recently proposed to use a stochastic differential equation (SDE) instead of an ODE. For the unconstrained case the use of SDEs was also advocated in [21,22,23,24]. The advantage of the SDE is that it can help the algorithm escape from local minima and eventually reach the global solution.…”

“…This function is usually referred to as the annealing schedule. In order for the algorithm to theoretically exhibit convergence to the global solution, the annealing schedule is selected as follows [21,23,22,14]:…”

Modelling and Estimation for Random Fields

“…To obtain Markov chain annealing algorithms we simply replace the fixed temperature T in the above Markov chain sampling methods by a temperature schedule {Tk} (where typically Tk --0). We can establish a weak convergence result for a nonstationary continuous (4.13) We remark that there has been a lot of work establishing convergence results for discrete state Markov chain annealing algorithms [6], [24]- [27], and also for the Markov diffusion annealing algorithm [7], [28], [29]. However, there are very few convergence results for continuous state Markov chain algorithms.…”

“…The constant Co) plays a critical role in the convergence of Xk as k --+ oo and also Y(t) as t -+ oo. In [28] it is shown that the constant C( (denoted there by co) has an interpretation in terms of the action functional for a family of perturbed dynamical systems; see [28] for a further discussion of C) including some examples. 2.…”

“…The asymptotic behavior of Y(t) as t --oo has been studied intensively by a number of researchers. In [7], [29] convergence results where obtained by considering a version of (4.17) with a reflecting boundary; in [28] the reflecting boundary was removed. Our analysis of {Xk} is based on the analysis of Y(t) developed in [28] where the following result is proved: if U(.)…”