Reflections on Output Analysis for Multistage Stochastic Linear Programs

Abstract

When solving a dynamic decision problem under uncertainty it is essential to choose or to build a suitable model taking into account the nature of the real-life problem, the character and availability of the input data, etc. There exist hints when to use stochastic dynamic programming models or multiperiod and multistage stochastic programs. Still, it is difficult to provide a general recipe. We refer to recent papers [1, 15] which characterize the main features and basic requirements of these models and indicate the cases which allow for multimodeling and comparisons or for exploitation of different approaches within one decision problem.

For both approaches, solution procedures are mostly based on an approximation scheme and it is important to relate the optimal value and optimal solutions of an approximating problem and the underlying one. It is interesting to recognize that methods of output analysis for stochastic dynamic programs were developed already in the eighties, cf. [25] and references ibidem. Regarding the solution method — the backward recursion connected with the principle of optimality — special emphasis was put on properties of discretization of state and control spaces.

We shall focus on multistage stochastic linear programs with recourse and with already given horizon and stages, that result by approximation of the underlying probability distribution. It turns out that generalization of various results well-known for two-stage stochastic linear programs to the multistage problems is not straightforward and it requires various additional assumptions, e.g., the interstage independence. We shall discuss possible generalizations of output analysis methods as delineated in [10].