Sampling-based decomposition methods for risk-averse multistage programs

Abstract

We define a risk averse nonanticipative feasible policy for multistage stochastic programsand propose a methodology to implement it. The approach is based on dynamic programmingequations written for a risk averse formulation of the problem.This formulation relies on a new class of multiperiod risk functionals called extended polyhedralrisk measures. Dual representations of such risk functionals are given and used to derive conditionsof coherence. In the one-period case, conditions for convexity and consistency with second orderstochastic dominance are also provided. The risk averse dynamic programming equations arespecialized considering convex combinations of one-period extended polyhedral risk measures suchas spectral risk measures.To implement the proposed policy, the approximation of the risk averse recourse functionsfor stochastic linear programs is discussed. In this context, we detail a stochastic dual dynamicprogramming algorithm which converges to the optimal value of the risk averse problem.