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We consider the problem of choosing design parameters to minimize theprobability of an undesired rare event that is described through the average of$n$ iid random variables. Since the probability of interest for near optimaldesign parameters is very small, one needs to develop suitable acceleratedMonte-Carlo methods for estimating the objective function of interest. One ofthe challenges in the study is that simulating from exponential twists of thelaws of the summands may be computationally demanding since these transformedlaws may be non-standard and intractable. We consider a setting where thesummands are given as a nonlinear functional of random variables that are moretractable for importance sampling in that the exponential twists of theirdistributions take a simpler form (than that for the original summands). Wealso study the closely related problem of estimating buffered probability ofexceedance and provide the first rigorous results that relate the asymptoticsof buffered probability and that of the ordinary probability under a largedeviation scaling. The analogous minimization problem for buffered probability,under conditions, can be formulated as a convex optimization problem whichmakes it more tractable than the original optimization problem. We show that,under conditions, changes of measures that are asymptotically efficient (underthe large deviation scaling) for estimating ordinary probability are alsoasymptotically efficient for estimating the buffered probability of exceedance.We embed the constructed importance sampling scheme in suitable gradientdescent/ascent algorithms for solving the optimization problems of interest.Implementation of schemes for some examples is illustrated throughcomputational experiments.

Minimization of a Class of Rare Event Probabilities and Buffer Probabilities of Exceedance