Buffered probability of exceedance: Mathematical properties and optimization^∗

This paper studies a probabilistic characteristic called buffered probability of exceedance (bPOE). It is a function of a random variable and a real-valued threshold. By definition, bPOE is the probability of a tail such that the average of this tail equals the threshold. This characteristic is an extension of the so-called buffered failure probability and it is equal to one minus inverse of the conditional value-at-risk (CVaR). bPOE is a quasi-convex function of the random variable w.r.t. the regular addition operation and a concave function w.r.t. the mixture operation; it is a monotonic function of the random variable; it is a strictly decreasing function of the threshold on the interval between the expectation and the essential supremum. The multiplicative inverse of the bPOE is a convex function of the threshold, and a piecewise-linear function in the case of discretely distributed random variable. The paper provides efficient calculation formulas for bPOE. Minimization of bPOE is reduced to a convex program for a convex feasible region and to linear programming for a polyhedral feasible region and discretely distributed random variables. A family of bPOE minimization problems and corresponding CVaR minimization problems share the same set of optimal solutions.