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Minimizing buffered probability of exceedance by progressive hedging

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Abstract

Stochastic programming problems have for a long time been posed in terms of minimizing the expected value of a random variable influenced by decision variables, but alternative objectives can also be considered, such as minimizing a measure of risk. Here something different is introduced: minimizing the buffered probability of exceedance for a specified loss threshold. The buffered version of the traditional concept of probability of exceedance has recently been developed with many attractive properties that are conducive to successful optimization, in contrast to the usual concept, which is often posed simply as the probability of failure. The main contribution here is to demonstrate that in minimizing buffered probability of exceedance the underlying convexities in a stochastic programming problem can be maintained and the progressive hedging algorithm can be employed to compute a solution.

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Notes

  1. 1.

    This represention goes back to [27] and was extended from stochastic programming to stochastic variational inequalities in [29].

  2. 2.

    An extension of progressive hedging to iterate also on such multipliers with extra proximal terms is available in [22].

  3. 3.

    The second part of this has already been addressed; the existence part can be handled by compactness of the sets \(C(\xi )\) or more generality some joint aspects of these sets and growth properties of the functions \(f(\cdot ,\xi )\). For more on these issues, see [28, §4].

  4. 4.

    http://www.ise.ufl.edu/uryasev/research/testproblems/financial_engineering/portfolio-optimization-cvar-vs-st_dev/.

  5. 5.

    Portfolio Safeguard (PSG), http://www.aorda.com.

  6. 6.

    http://www.ise.ufl.edu/uryasev/research/testproblems/financial_engineering/case-study-cash-matching-with-bpoe-and-cvar-functions/.

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Acknowledgements

This research was sponsored for both authors by DARPA EQUiPS Grant SNL 014150709. The authors are grateful also to Dr. Viktor Kuzmeno for help with conducting the numerical case study.

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Correspondence to R. Tyrrell Rockafellar.

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Rockafellar, R.T., Uryasev, S. Minimizing buffered probability of exceedance by progressive hedging. Math. Program. (2020) doi:10.1007/s10107-019-01462-4

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Keywords

  • Convex stochastic programming problems
  • Probability of failure
  • Probability of exceedance
  • Buffered probability of failure
  • Buffered probability of exceedance
  • Quantiles
  • Superquantiles
  • Conditional value-at-risk
  • Progressive hedging algorithm

Mathematics Subject Classification

  • 90C15