On Solving Multistage Stochastic Programs with Coherent Risk Measures

Abstract

We consider a class of multistage stochastic linear programs in which at each stage a coherent risk measure of future costs is to be minimized. A general computational approach based on dynamic programming is derived that can be shown to converge to an optimal policy. By computing an inner approximation to future cost functions, we can evaluate an upper bound on the cost of an optimal policy, and an outer approximation delivers a lower bound. The approach we describe is particularly useful in sampling-based algorithms, and a numerical example is provided to show the e¢ cacy of the methodology when used in conjunction with stochastic dual dynamic programming.

Equilibrium, uncertainty and risk in hydro-thermal electricity systems

“…However we show that given this information, the planner can in principle solve a risk-averse dynamic optimization problem with an appropriate coherent risk measure (using e.g. the methods discussed in [13]) to yield a stochastic process of energy prices that correspond to the outcomes of a competitive equilibrium with risk trading.…”

“…Note that using the same argument as in Theorem 3, we may assume without loss of generality that µ * (m) > 0,m ∈ N \ {0} so we can recover (13). This shows that a∈A θ a (n) = θ s (n) for every n with depth δ L − 1.…”

“…This shows that a∈A θ a (n) = θ s (n) for every n with depth δ L − 1. By induction, Lemma 8 implies for any node n ∈ N \ L that the risk price µ * that solves DTP(n) gives an optimal solution (θ a (n) , W a (m)) to RT(a, n) for each agent a ∈ A, and satisfies (13). Now observe that by Assumption 4, given µ ∈ D s (n), we can construct λ satisfying…”

Risk aversion in multistage stochastic programming: A modeling and algorithmic perspective

“…A number of recent papers have considered the importance of measuring risk in MSSPs (see, for instance, Eichhorn and Römisch 2005;Pflug and Römisch 2007;Collado et al 2012;Shapiro 2012a;Philpott and de Matos 2012;Philpott et al 2013;Shapiro et al 2013;Kozmık and Morton 2013;Pagnoncelli and Piazza 2012;Guigues and Sagastizábal 2013;Pflug and Pichler 2014b). Several of these papers focus on how to adapt existing algorithms from the risk-neutral case to the risk-averse case, often with the Conditional Value-at-Risk as the risk measure.…”

“…,ξ t−1 . Such nested risk measures have been object of extensive study, see for instance (2012) and Philpott et al (2013).…”