Scenario decomposition of risk-averse stochastic optimization problems
In the last decade the theory of coherent risk measures established itself as an alternative to expected utility models of risk averse preferences in stochastic optimization. Recently, increased attention is paid to dynamic measures of risk, which allow for risk-averse evaluation of streams of costs or rewards. When used in stochastic optimization models, dynamic risk measures lead to a new class of problems, which are significantly more complex than their risk-neutral counterparts.
... rparts. Decomposition, an established and efficient approach to risk-neutral multistage stochastic optimization problems, cannot be directly applied to risk-averse models. With dynamic risk measures, the main feature facilitating decomposition, the integral form of the objective function, is absent. Our main objective is to overcome this difficulty by exploiting specific structure of dynamic risk measures, and to develop new decomposition methods that extend the ideas of earlier approaches to risk-neutral problems. In this work we develop generalizations of scenario decomposition methods, in the spirit of J.M. Mulvey and A. Ruszczynski, "A new scenario decomposition method for large-scale stochastic optimization'" Operations Research 43, 1995. The key to success is the use of dual properties of dynamic measures of risk to construct a family of risk-neutral approximations of the problem. First, we define and analyze a two-stage risk-averse stochastic optimization problem. Next, we develop methods to solve efficiently this problem. Later, we formally define a multistage risk-averse stochastic optimization problem and we discuss its properties. We also develop efficient methods to solve the multistage problem and apply these to an inventory planning and assembly problem. Finally, we analyze and compare the results of our computational experiments.