Cross-Entropy method: convergence issues for extended implementation

The cross-entropy method (CE) developed by R. Rubinstein is an elegant practical principle for simulating rare events. The method approximates the probability of the rare event by means of a family of probabilistic models. The method has been extended to optimization, by considering an optimal event as a rare event. CE works rather good when dealing with deterministic function optimization. Now, it appears that two conditions are needed for a good convergence of the method. First, it is necessary to have a family of models sufficiently flexible for discriminating the optimal events. Indirectly, it appears also that the function to be optimized should be deterministic. The purpose of this paper is to consider the case of partially discriminating model family, and of stochastic functions. It will be shown on simple examples that the CE could fail when relaxing these hypotheses. Alternative improvements of the CE method are investigated and compared on random examples in order to handle this issue.