Sampling-based approximation algorithms for multistage stochastic optimization

摘要

Stochastic optimization problems provide a means to model uncertainty in the input data where the uncertainty is modeled by a probability distribution over the possible realizations of the actual data. We consider a broad class of these problems in which the realized input is revealed through a series of stages and hence are called multistage stochastic programming problems. Multistage stochastic programming and, in particular, multistage stochastic linear programs with full recourse, is a domain that has received a great deal of attention within the operations research community, mostly from the perspective of computational results in application settings. Our main result is to give the first fully polynomial approximation scheme for a broad class of multistage stochastic linear programming problems with any constant number of stages. The algorithm analyzed, known as the sample average approximation method, is quite simple and is the one most commonly used in practice. The algorithm accesses the input by means of a "black box" that can generate, given a series of outcomes for the initial stages, a sample of the input according to the conditional probability distribution (given those outcomes). We use this to obtain the first approximation algorithms for a variety of k-stage generalizations of basic combinatorial optimization problems including the set cover, vertex cover, multicut on trees, facility location, and multicommodity flow problems.