Moment calculations for piecewise-defined functions: an application to stochastic optimization with coherent risk measures (Q993722)

The authors present a new methodology for supporting the formulation of stochastic programming (SP) problems involving piecewise-defined (PD) functions which widely present in O.R. applications, in particular, in all (SP) problems with coherent measures of risk. They introduce a new definition of multivariate PD functions and derive formal results for their continuity and differentiability. The authors introduce the notion of distinct constituents, obtain closed-form expressions for the computation of the moment of PD functions and utilize it to derive the analytic and stochastic properties of \(\max \left( \cdot \right) \) and \(\left| \cdot \right| \) PD functions which appear frequently in O.R. applications. Finally, they derive results for SP problems with mean-deviation, mean-upper-semideviation and conditional value at risk (CVAR).