  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.
