Evaluation of scenario-generation methods for stochastic programming

Stochastic programs can only be solved with discrete distributions of limited cardinality. Input, however, normally comes in the form of continuous distributions or large data sets. Creating a limited discrete distribution from input is called scenario generation. In this paper, we discuss how to evaluate the quality or suitability of scenario generation methods for a given stochastic programming model. We formulate minimal requirements that should be imposed on a scenario generation method before it can be used for solving the stochastic programming model. We also show how the requirements can be tested. The procedures for testing a scenario generation method is illustrated on a case from portfolio management.

[1]  C. D. Vale,et al.  Simulating multivariate nonnormal distributions , 1983 .

[2]  R. Wets,et al.  Stochastic programming , 1989 .

[3]  G. Dantzig,et al.  Large-Scale Stochastic Linear Programs: Importance Sampling and Benders Decomposition , 1991 .

[4]  Julia L. Higle,et al.  Stochastic Decomposition: An Algorithm for Two-Stage Linear Programs with Recourse , 1991, Math. Oper. Res..

[5]  Yuri M. Ermoliev,et al.  Stochastic quasigradient methods for optimization of discrete event systems , 1992, Ann. Oper. Res..

[6]  G. Infanger,et al.  Planning under uncertainty solving large-scale stochastic linear programs , 1992 .

[7]  Gerd Infanger,et al.  Monte Carlo (importance) sampling within a benders decomposition algorithm for stochastic linear programs , 1991, Ann. Oper. Res..

[8]  James E. Smith.  Moment Methods for Decision Analysis , 1993 .

[9]  W. Ziemba,et al.  The Russell-Yasuda Kasai Model: An Asset/Liability Model for a Japanese Insurance Company Using Multistage Stochastic Programming , 1994 .

[10]  John R. Birge,et al.  Introduction to Stochastic Programming , 1997 .

[11]  Michael A. H. Dempster,et al.  Evpi-Based Importance Sampling Solution Procedures for Multistage Stochastic Linear Programmes on Parallel Mimd Architectures , 1997 .

[12]  Mico Loretan,et al.  Generating market risk scenarios using principal components analysis: methodological and practical considerations , 1997 .

[13]  Philip M. Lurie,et al.  An Approximate Method for Sampling Correlated Random Variables From Partially-Specified Distributions , 1998 .

[14]  R. Nelsen.  An Introduction to Copulas , 1998 .

[15]  R. Clemen,et al.  Correlations and Copulas for Decision and Risk Analysis , 1999 .

[16]  David P. Morton,et al.  Monte Carlo bounding techniques for determining solution quality in stochastic programs , 1999, Oper. Res. Lett..

[17]  Barry L. Nelson,et al.  Modeling and Generating Multivariate Time Series with Arbitrary Marginals Using a Vector Autoregress , 2000 .

[18]  Jitka Dupacová,et al.  Scenarios for Multistage Stochastic Programs , 2000, Ann. Oper. Res..

[19]  J. Lyhagen.  A method to generate multivariate data with moments arbitrary close to the desired moments , 2001 .

[20]  Stein W. Wallace,et al.  Generating Scenario Trees for Multistage Decision Problems , 2001, Manag. Sci..

[21]  Georg Ch. Pflug,et al.  Scenario tree generation for multiperiod financial optimization by optimal discretization , 2001, Math. Program..

[22]  Roy Kouwenberg,et al.  Scenario generation and stochastic programming models for asset liability management , 2001, Eur. J. Oper. Res..

[23]  Teemu Pennanen,et al.  Integration quadratures in discretization of stochastic programs , 2002 .

[24]  S. Zenios,et al.  CVaR models with selective hedging for international asset allocation , 2002 .

[25]  Stein-Erik Fleten,et al.  The performance of stochastic dynamic and fixed mix portfolio models , 2002, Eur. J. Oper. Res..

[26]  Roger Halldin.  Scenario trees for inflow modelling in stochastic optimisation for energy planning , 2002 .

[27]  Werner Römisch,et al.  Scenario Reduction Algorithms in Stochastic Programming , 2003, Comput. Optim. Appl..

[28]  Jitka Dupacová,et al.  Scenario reduction in stochastic programming , 2003, Math. Program..

[29]  Michal Kaut,et al.  Stability analysis of a portfolio management model based on the conditional value-at-risk measure , 2003 .

[30]  Michal Kaut,et al.  A Heuristic for Moment-Matching Scenario Generation , 2003, Comput. Optim. Appl..

[31]  D. Morton,et al.  Assessing policy quality in multi-stage stochastic programming , 2004 .

[32]  Suvrajeet Sen,et al.  The Scenario Generation Algorithm for Multistage Stochastic Linear Programming , 2005, Math. Oper. Res..

[33]  David P. Morton,et al.  Assessing solution quality in stochastic programs , 2006, Algorithms for Optimization with Incomplete Information.

[34]  Alexander Shapiro,et al.  The empirical behavior of sampling methods for stochastic programming , 2006, Ann. Oper. Res..

[35]  Ronald Hochreiter,et al.  Financial scenario generation for stochastic multi-stage decision processes as facility location problems , 2007, Ann. Oper. Res..

[36]  Antonio Alonso Ayuso,et al.  Introduction to Stochastic Programming , 2009 .