Cross Entropy Methods

The Cross Entropy Method

The cross-entropy (CE) method [56] is a general Monte Carlo approach to combinatorial and continuous multi-extremal optimization [195]. The method originated from the field of rare event simulation, where very small probabilities need to be accurately estimated.

Consider a Monte Carlo simulation which draws instances from the true distribution of events. Such a problem usually requires an inordinate number of draws before enough of the rare events are seen to make a reliable estimate of their probability of occurring. A better way is to use Importance Sampling (IS) [7, 66, 118] which is a general technique for estimating the properties of a particular distribution, while only having samples generated from a different distribution than the distribution of interest. However, such a method has the drawback that the optimal reference parameters to be used in IS are usually difficult to obtain. The cross entropy method thus has the advantage that it provides a simple adaptive procedure for estimating the optimal reference parameters with asymptotic convergence properties. Not only being applied to the estimates of rare-event problems in dynamic models [57, 140], a simple modification of the CE method [195] has been also applied to solve difficult combinatorial optimization problems (COPs), see (([193, 194]).