A Primal-Dual Interior Point and Decomposition Approach to Multi-Stage Stochastic Programming

How to make decisions while the future is full of uncertainties is a major problem shared virtually by every human being including housewives, firm managers, as well as politicians. In this paper we introduce a mathematical programming resolution to the problem, namely multi-stage stochastic programming. An advantage of that approach is obviously that we will be working with a precise, tangible, and theoretically computable model. The difficulty, on the other hand, is that the size of the model quickly becomes gigantic, making the problem intractable unless the structure of the model is carefully exploited. In this paper we discuss a particular decomposition approach to solve multi-stage stochastic linear programming, based on a high performance interior point method. Numerical experiences suggest that the method requires only a few number of iteration steps to reach a high precision solution, almost regardless the size of the problem. Attention is thus paid to solving a large size Newton equation needed for obtaining a search direction at each iteration by decomposition. It turns out that it is possible to reduce the computational effort to a minimum level, i.e., the number of elementary operations at each iteration is linear in the total number of scenarios in the original multi-stage stochastic linear programming model.