Optimization Letters

, Volume 13, Issue 6, pp 1223–1237 | Cite as

Higher-moment buffered probability

  • D. P. KouriEmail author
Original Paper


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.


Risk measures Chance constraints Stochastic programming Conditional value-at-risk 



The author gratefully acknowledges the comments and suggestions from the reviewers. Their comments helped to simplify the original results in Sect. 3. This research was sponsored by DARPA EQUiPS Grant SNL 014150709


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Optimization and Uncertainty QuantificationSandia National LaboratoriesAlbuquerqueUSA

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