Sampling Within Stochastic Linear Programming

Abstract

We are faced with the task of solving two stage stochastic linear programs with recourse (SLP), such as those discussed in Chapter 1. As noted in Theorem 1.1, this class of problems are convex programs and in principle, any of a number of convex programming algorithms (e.g. subgradient methods, cutting plane methods, Lagrangian based methods etc..) can be used to solve SLPs. From the discussions in Chapter 1, it is clear that in most cases, the stochastic nature of the problem precludes the precise determination of subgradients and objective function values. When the presence of random variables prevents the precise determination of such quantities, it is natural to use random samples in the development of statistical estimates of these quantities. Until recently, only subgradient methods were incorporated within a sampling framework, and the resulting methods became known as stochastic quasi-gradient (SQG) methods (see Er-moliev [1988] for a survey). While SQG methods are applicable to very general stochastic convex programs, they suffer from many of the drawbacks of deterministic subgradient methods. In particular, the choice of effective steplengths is often problem dependent. In addition, the incorporation of optimality criteria within these algorithms remains elusive. Nevertheless, because of the incorporation of sampling within the algorithm, SQG methods are able to address SLPs with a large number of outcomes, as well as problems with continous random variables. In developing the Stochastic Decomposition (SD) method, our goal is to bestow these advantages on cutting plane algorithms, which have remained the mainstay for SLPs for several decades (Van Slyke and Wets [1969], Ruszczyński [1986], Birge and Louveaux [1988], Gassmann [1990] etc). The randomization of cutting plane methods has provided the capability of solving truly large scale stochastic programs such as SSN and STORM presented in Chapter 1.