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Title: 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.

Authors:
ORCiD logo [1]
  1. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Publication Date:
Research Org.:
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Sponsoring Org.:
Defense Advanced Research Projects Agency (DARPA); USDOE
OSTI Identifier:
1497635
Report Number(s):
SAND-2019-2008J
Journal ID: ISSN 1862-4472; 672842
Grant/Contract Number:  
AC04-94AL85000
Resource Type:
Accepted Manuscript
Journal Name:
Optimization Letters
Additional Journal Information:
Journal Name: Optimization Letters; Journal ID: ISSN 1862-4472
Publisher:
Springer Nature
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; Risk measures; Chance constraints; Stochastic programming; Conditional value-at-risk

Citation Formats

Kouri, D. P. Higher-moment buffered probability. United States: N. p., 2019. Web. doi:10.1007/s11590-018-1359-2.
Kouri, D. P. Higher-moment buffered probability. United States. doi:10.1007/s11590-018-1359-2.
Kouri, D. P. Fri . "Higher-moment buffered probability". United States. doi:10.1007/s11590-018-1359-2. https://www.osti.gov/servlets/purl/1497635.
@article{osti_1497635,
title = {Higher-moment buffered probability},
author = {Kouri, D. P.},
abstractNote = {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.},
doi = {10.1007/s11590-018-1359-2},
journal = {Optimization Letters},
number = ,
volume = ,
place = {United States},
year = {2019},
month = {2}
}

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    Works referencing / citing this record:

    On NEMYTSKIJ Operators inLp-Spaces of Abstract Functions
    journal, January 1992

    • Goldberg, H.; Kampowsky, W.; Tröltzsch, F.
    • Mathematische Nachrichten, Vol. 155, Issue 1
    • DOI: 10.1002/mana.19921550110

    Filters in topology optimization based on Helmholtz-type differential equations
    journal, December 2010

    • Lazarov, B. S.; Sigmund, O.
    • International Journal for Numerical Methods in Engineering, Vol. 86, Issue 6
    • DOI: 10.1002/nme.3072

    Kusuoka representation of higher order dual risk measures
    journal, May 2010

    • Dentcheva, Darinka; Penev, Spiridon; Ruszczyński, Andrzej
    • Annals of Operations Research, Vol. 181, Issue 1
    • DOI: 10.1007/s10479-010-0747-5

    Cash flow matching with risks controlled by buffered probability of exceedance and conditional value-at-risk
    journal, November 2016

    • Shang, Danjue; Kuzmenko, Victor; Uryasev, Stan
    • Annals of Operations Research, Vol. 260, Issue 1-2
    • DOI: 10.1007/s10479-016-2354-6

    On buffered failure probability in design and optimization of structures
    journal, May 2010


    Higher moment coherent risk measures
    journal, August 2007


    Coherent Measures of Risk
    journal, July 1999

    • Artzner, Philippe; Delbaen, Freddy; Eber, Jean-Marc
    • Mathematical Finance, Vol. 9, Issue 3
    • DOI: 10.1111/1467-9965.00068

    Risk Measures on Orlicz Hearts
    journal, April 2009


    Convex Approximations of Chance Constrained Programs
    journal, January 2007

    • Nemirovski, Arkadi; Shapiro, Alexander
    • SIAM Journal on Optimization, Vol. 17, Issue 4
    • DOI: 10.1137/050622328

    Inexact Objective Function Evaluations in a Trust-Region Algorithm for PDE-Constrained Optimization under Uncertainty
    journal, January 2014

    • Kouri, D. P.; Heinkenschloss, M.; Ridzal, D.
    • SIAM Journal on Scientific Computing, Vol. 36, Issue 6
    • DOI: 10.1137/140955665

    A multi-fidelity framework for investigating the performance of super-cavitating hydrofoils under uncertain flow conditions
    conference, January 2017

    • Bonfiglio, Luca; Perdikaris, Paris; Brizzolara, Stefano
    • 19th AIAA Non-Deterministic Approaches Conference
    • DOI: 10.2514/6.2017-1328