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Sampling-based decomposition methods for multistage stochastic programs based on extended polyhedral risk measures

Vincent Guigues (vguigues***at***impa.br)
Werner Romisch (romisch***at***math.hu-berlin.de)

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

Keywords: Convex risk measure; Coherent risk measure; Stochastic programming; Risk averse optimization; Decomposition algorithms; Monte-Carlo sampling; Spectral risk measure; CVaR

Category 1: Stochastic Programming


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Entry Submitted: 10/12/2010
Entry Accepted: 10/13/2010
Entry Last Modified: 08/21/2012

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