An exploration of stochastic decomposition algorithms for stochastic linear programs with recourse

Abstract

Stochastic linear programs are linear programs in which some of the problem data are random variables. The particular kind of programs that we study belong to the recourse model. Under this model, some decisions are postponed until better information becomes available (e.g., an outcome of a random variable is realized), while other decisions must be made 'here and now.' For example, in a telecommunication network planning problem, decisions regarding the addition of network capacity have to be made before knowing customer demand (i.e., 'here and now'). Once the demand is realized, efficient usage of the network can then be determined. This work explores algorithms for the solution of such programs: stochastic linear programs with recourse. The algorithms investigated can be described as decomposition based cutting plane methods in which the cuts are estimated from random samples. Moreover, the algorithms all use the incremental sampling plan inherent to the Stochastic Decomposition (SD) algorithm developed by Higle and Sen in 1991. Our study includes both two stage and multistage programs. For the solution of two stage programs, we present the Conditional Stochastic Decomposition (CSD) algorithm, a multicut version of the SD algorithm. CSD is most suitable for situations in which data are difficult to obtain and may be computationally intense. Because of this potential intensity, we explore algorithms which require less computational effort than CSD. These algorithms combine features of both CSD and SD and are referred to as hybrid algorithms. Following our exploration of these algorithms for two stage problems, we next explore an extension of the SD algorithm that can be used for multistage problems with stagewise independent random variables. For the sake of notational brevity, our technical development is centered around the three stage case, although the extension to multistage problems is straightforward. Under mild conditions, convergence results similar to those found in the two stage algorithms hold. Multistage stochastic decomposition is currently a largely uncharted area. Our research represents the first major effort in this direction.