2009-05-19Buch DOI: 10.18452/3026
A computational study of a solver system for processing two-stage stochastic linear programming problems
Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, Institut für Mathematik
Formulation of stochastic optimisation problems and computational algorithms for their solution continue to make steady progress as can be seen from an analysis of many developments in this ﬁeld. The edited volume by Wallace and Ziemba (2005) outlines both the SP modelling systems and many applications in diverse domains. More recently, Fabozzi (2008) has considered the application of SP models to challenging ﬁnancial engineering problems. The tightly knit yet highly focused group of researchers COSP: Committee on Stochastic Programming, their triennial international SP conference, and their active website points to the progressive acceptance of SP as a valuable decision tool. At the same time many of the major software vendors, namely, XPRESS, AIMMS, and MAXIMAL GAMS have started offering SP extensions to their optimisation suites. Our analysis of the modelling and algorithmic solver requirements reveals that (a) modelling support (b) scenario generation and (c) solution methods are three important aspects of a working SP system. Our research is focussed on all three aspects and we refer the readers to Valente et al. (2009) for modelling and Mitra et al. (2007) and Di Domenica et al. (2009) for scenario generation. In this paper we are concerned entirely with com- putational solution methods. Given the tremendous advance in LP solver algorithms there is certain amount of complacency that by constructing a ”deterministic equivalent” problems it is possible to process most realistic instances of SP problems. In this paper we highlight the shortcoming of this line of argument. We describe the implementation and reﬁnement of established algorithmic methods and report a computational study which clearly underpins the superior scale up properties of the solution methods which are described in this paper. A taxonomy of the important class of SP problems may be found in Valente et al. (2008, 2009). The most important class of problems with many applications is the two-stage stochastic programming model with recourse, which originated from the early research of Dantzig (1955), Beale (1955) and Wets (1974). A comprehensive treatment of the model and solution methods can be found in Kall and Wallace (1994), Prekopa (1995), Birge and Louveaux (1997), Mayer (1998), Ruszczynski and Shapiro (2003), and Kall and Mayer (2005). Some of these monographs contain generalisations of the original model. Colombo et al. (2006) and Gassmann and Wallace (1996) describe computational studies which are based on interior point method and simplex based methods respectively. The rest of this paper is organised in the following way. In section 2 we introduce the model setting of the two stage stochastic programming problem, in section 3 we consider a selection of solution methods for processing this class of problems. The established approaches of processing the deterministic equivalent LP form, the decomposition approach of Benders, the need for regularisation are also discussed. We also introduce the concept of level decomposition and explain how it ﬁts into the concept of regularisation. In section 4 we set out the computational study and in section 5 we summarise our conclusions.
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