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Moment calculations for piecewise-defined functions: an application to stochastic optimization with coherent risk measures

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Abstract

This work introduces a new analytical approach to the formulation of optimization problems with piecewise-defined (PD) objective functions. First, we introduce a new definition of multivariate PD functions and derive formal results for their continuity and differentiability. Then, we obtain closed-form expressions for the calculation of their moments. We apply these findings to three classes of optimization problems involving coherent risk measures. The method enables one to obtain insights on problem structure and on sensitivity to imprecision at the problem formulation stage, eliminating reliance on ad-hoc post-optimality numerical calculations.

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Authors and Affiliations

  1. ELEUSI Research Center and Department of Decision Sciences, Bocconi University, Via Roentgen 1, 20136, Milan, Italy

    E. Borgonovo & L. Peccati

Authors
  1. E. Borgonovo
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  2. L. Peccati
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Correspondence to L. Peccati.

Additional information

The authors wish to thank the anonymous referees for the very insightful comments, that have greatly contributed in the final realization of the present paper. A special thank to Jacques Carette for the very perceptive and useful suggestions on an earlier version of the present paper. Financial support from the ELEUSI Research Center of Bocconi University is gratefully acknowledged.

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Borgonovo, E., Peccati, L. Moment calculations for piecewise-defined functions: an application to stochastic optimization with coherent risk measures. Ann Oper Res 176, 235–258 (2010). https://doi.org/10.1007/s10479-008-0504-1

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