Higher-moment buffered probability

Abstract

In stochastic optimization, probabilities naturally arise as cost functionals and chance constraints. Unfortunately, these functions are difficult to handle both theoretically and computationally. The buffered probability of failure and its subsequent extensions were developed as numerically tractable, conservative surrogates for probabilistic computations. In this manuscript, we introduce the higher-moment buffered probability. Whereas the buffered probability is defined using the conditional value-at-risk, the higher-moment buffered probability is defined using higher-moment coherent risk measures. In this way, the higher-moment buffered probability encodes information about the magnitude of tail moments, not simply the tail average. We prove that the higher-moment buffered probability is closed, monotonic, quasi-convex and can be computed by solving a smooth one-dimensional convex optimization problem. These properties enable smooth reformulations of both higher-moment buffered probability cost functionals and constraints.