Improving convergence of the stochastic decomposition algorithm by using an efficient sampling technique

Abstract

This work focuses on the basic stochastic decomposition (SD) algorithm of Higle and Sen [J.L. Higle, S. Sen, Stochastic Decomposition, Kluwer Academic Publishers, 1996] for two-stage stochastic linear programming problems with complete recourse. The algorithm uses sampling when the random variables are represented by continuous distribution functions. Traditionally, this method has been applied by using Monte Carlo (MC) sampling to generate the samples of the stochastic variables. However, Monte Carlo methods can result in large error bounds and variance. Hence, some other approaches use importance sampling to reduce variance and achieving convergence faster that the method based on the Monte Carlo sampling technique. This work proposes an improvement on this respect. Hence, we propose to replace the use of the Monte Carlo sampling technique or the importance sampling in the SD algorithm by the use of the novel Hammersley sequence sampling (HSS) technique. Recently, such a technique has proved to provide better uniformity properties than other sampling techniques and, as a consequence, the variance and the number of samples required for convergence are reduced. Also, we use a fractal dimension approach to characterize the error of the estimation of the recourse function based on sampling. The computational implementation of the algorithm involves a framework that integrates the GAMS modeling environment, the HSS sampling code (FORTRAN) and a C++ program which generates appropriate LP problems for each SD iteration. The algorithm has been tested with several case studies representing chemical engineering applications and the results are discussed.