Polyhedral Risk Measures in Stochastic Programming

Abstract

We consider stochastic programs with risk measures in the objective and study stability properties as well as decomposition structures. Thereby we place emphasis on dynamic models, i.e., multistage stochastic programs with multiperiod risk measures. In this context, we define the class of polyhedral risk measures such that stochastic programs with risk measures taken from this class have favorable properties. Polyhedral risk measures are defined as optimal values of certain linear stochastic programs where the arguments of the risk measure appear on the right-hand side of the dynamic constraints. Dual representations for polyhedral risk measures are derived and used to deduce criteria for convexity and coherence. As examples of polyhedral risk measures we propose multiperiod extensions of the Conditional-Value-at-Risk.

Optimization of Dispersed Energy Supply —Stochastic Programming with Recombining Scenario Trees

“…For this purpose, especially the class of polyhedral risk measures [5] seems to be useful, since the linear structure of the optimization problem is maintained.…”

Gain-loss pricing under ambiguity of measure

“…Here, in the notation of [12] we have D 1 = R and D 2 is the interval [1, λ] by a simple computation. Then doing calculations similar to those in Example 2.10 of [12], we have that the dual representation (2.3) of Theorem 2.4 of [12] holds.…”

“…We obtain our results in an infinite state probability setting, and using ideas from stochastic minimax programming [28] and from polyhedral risk measures [12]. For an in-depth treatment of risk measures in finance under discrete-time models, we direct the reader to the book by Föllmer and Schied [16], the recent paper by Ben-Tal and Teboulle [2], and to the papers by Ruszczyński and Shapiro [25,26].…”

Stability and Scenario Trees for Multistage Stochastic Programs

“…Typically, risk functionals are inherently nonlinear. If, however, a multiperiod polyhedral risk functional [11] replaces the expectation in (1), the resulting risk-averse stochastic program may be reformulated as a linear multistage stochastic program of the form (1) by introducing new state variables and (linear) constraints (see [11,Section 4]). Moreover, it is shown in [12] that the stability behavior of the reformulation does not change (when compared with the original problem with expectation objective) if the multiperiod polyhedral (convex) risk functional has bounded L 1 -level sets.…”