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

We consider the problem of choosing design parameters to minimize the probability of an undesired rare event that is described through the average of $n$ iid random variables. Since the probability of interest for near optimal design parameters is very small, one needs to develop suitable accelerated Monte-Carlo methods for estimating the objective function of interest. One of the challenges in the study is that simulating from exponential twists of the laws of the summands may be computationally demanding since these transformed laws may be non-standard and intractable. We consider a setting where the summands are given as a nonlinear functional of random variables that are more tractable for importance sampling in that the exponential twists of their distributions take a simpler form (than that for the original summands). We also study the closely related problem of estimating buffered probability of exceedance and provide the first rigorous results that relate the asymptotics of buffered probability and that of the ordinary probability under a large deviation scaling. The analogous minimization problem for buffered probability, under conditions, can be formulated as a convex optimization problem which makes it more tractable than the original optimization problem. We show that, under conditions, changes of measures that are asymptotically efficient (under the large deviation scaling) for estimating ordinary probability are also asymptotically efficient for estimating the buffered probability of exceedance. We embed the constructed importance sampling scheme in suitable gradient descent/ascent algorithms for solving the optimization problems of interest. Implementation of schemes for some examples is illustrated through computational experiments.