Data-driven optimization of reliability using buffered failure probability

Gradients and Subgradients of Buffered Failure Probability

“…We consider two applications of Theorem 2.1. The first one discusses sensitivity analysis of a simple network and illustrates how the formula in [2] may not apply; see (2.2). The second one presents an optimality condition for constrained minimization of the buffered failure probability.…”

“…, ξ 16 , but g(ξ, x) has only three outcomes 1 − 2x, 1 − x and 1. This means that the formula (2.2) from [2] does not apply; it remains inapplicable after a consolidation of the probability space into one with only three outcomes.…”

“…. , ν} are all distinct, certain expectations are bounded away from zero and g(ξ i , • ) is smooth (i.e., continuously differentiable) for all i in a neighborhood of x; see Example 1 and [2]. Here, we derive a necessary condition for subgradients 2 of p at x without requiring distinct values of {g(ξ i , x), i = 1, .…”

“…Buffered failure probabilities and the closely related buffered probabilities of exceedance [4] appear in several optimization models, often leveraging their connection with superquantiles (a.k.a. conditional/average value-atrisk), with applications in reliability engineering [10,1,21,2,3], data analytics [8], finance [16] and to address natural disasters [4]. However, gradient expressions for buffered failure probabilities as functions of parameters are not fully developed.…”

“…However, engineering systems often involve nonconvex quantities of interest; see for example [1]. When a finite distribution produces distinct outcomes for the quantity of interest and certain expectations are bounded away from zero, gradient expressions for buffered failure probabilities become available [2]. We extend on these developments by considering both finite and continuous distributions, and circumventing any assumption about convexity.…”