Two-stage stochastic linear programming: Stochastic decomposition approaches.

Abstract

Stochastic linear programming problems are linear programming problems for which one or more data elements are described by random variables. Two-stage stochastic linear programming problems are problems in which a first stage decision is made before the random variables are observed. A second stage, or recourse decision, which varies with these observations compensates for any deficiencies which result from the earlier decision. Many applications areas including water resources, industrial management, economics and finance lead to two-stage stochastic linear programs with recourse. In this dissertation, two algorithms for solving stochastic linear programming problems with recourse are developed and tested. The first is referred to as Quadratic Stochastic Decomposition (QSD). This algorithm is an enhanced version of the Stochastic Decomposition (SD) algorithm of Higle and Sen (1988). The enhancements were designed to increase the computational efficiency of the SD algorithm by introducing a quadratic proximal term in the master program objective function and altering the manner in which the recourse function approximations are updated. We show that every accumulation point of an easily identifiable subsequence of points generated by the algorithm are optimal solutions to the stochastic program with probability 1. The various combinations of the enhancements are empirically investigated in a computational experiment using operations research problems from the literature. The second algorithm is an SD based algorithm for solving a stochastic linear program in which the recourse problem appears in the constraint set. This algorithm involves the use of an exact penalty function in the master program. We find that under certain conditions every accumulation point of a sequence of points generated by the algorithm is an optimal solution to the recourse constrained stochastic program, with probability 1. This algorithm is tested on several operations research problems.