The Sample Average Approximation Method for 2-stage Stochastic Optimization

We consider the Sample Average Approximation (SAA) method for 2-stage stochastic optimization problems with recourse and prove a polynomial time convergence theorem for the SAA method. In the 2-stage recourse model, where one makes decisions in two steps. First, given only distributional information about (some of) the data, one commits on initial (first-stage) actions, and then once the actual data is realized, according to the distribution, further recourse actions can be taken, so that one can augment the earlier solution to satisfy the revealed requirements, if necessary. Typically the recourse actions entail making decisions in rapid reaction to the observed scenario, that is, at the “last minute,” and are therefore costlier than decisions made ahead of time. The goal is to choose the first stage elements so as to minimize the sum of the cost incurred in the first stage and the expected cost incurred in the second stage, where the expectation is taken over all problem instances and these instances are distributed according to the given probability distribution. More formally, given a probability distribution on scenarios A and a vector x describing the first stage decisions, the cost incurred is given by h(x) = c(x) + EA [ fA(x, rA) ] where vector rA denotes the second stage decisions that are taken when scenario A materializes, c(x) is the cost incurred in the first stage, and fA(x, rA) is the cost of augmenting x to obtain the solution (x, rA) for scenario A. We want to choose x that minimizes the total cost h(x). Consider a discrete distribution and let pA denote the probability of scenario A. Then the objective function is h(x) = c(x) + ∑ A∈A pAfA(x, rA), where A denotes the set of all scenarios.