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Scenario tree reduction for multistage stochastic programs

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A framework for the reduction of scenario trees as inputs of (linear) multistage stochastic programs is provided such that optimal values and approximate solution sets remain close to each other. The argument is based on upper bounds of the L r -distance and the filtration distance, and on quantitative stability results for multistage stochastic programs. The important difference from scenario reduction in two-stage models consists in incorporating the filtration distance. An algorithm is presented for selecting and removing nodes of a scenario tree such that a prescribed error tolerance is met. Some numerical experience is reported.

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  • Casey M, Sen S (2005) The scenario generation algorithm for multistage stochastic linear programming. Math Oper Res 30: 615–631

    Article  Google Scholar 

  • Dupačová J, Consigli G, Wallace SW (2000) Scenarios for multistage stochastic programs. Ann Oper Res 100: 25–53

    Article  Google Scholar 

  • Dupačová J, Gröwe-Kuska N, Römisch W (2003) Scenario reduction in stochastic programming: an approach using probability metrics. Math Program 95: 493–511

    Article  Google Scholar 

  • Eichhorn A, Römisch W (2005) Polyhedral risk measures in stochastic programming. SIAM J Optim 16: 69–95

    Article  Google Scholar 

  • Eichhorn A, Römisch W (2006) Mean-risk optimization models for electricity portfolio management. In: Proceedings of PMAPS 2006 (Probabilistic Methods Applied to Power Systems), Stockholm, Sweden

  • Eichhorn A, Römisch W (2008) Stability of multistage stochastic programs incorporating polyhedral risk measures. Optimization 57: 295–318

    Article  Google Scholar 

  • Heitsch H, Römisch W (2003) Scenario reduction algorithms in stochastic programming. Comp Optim Appl 24: 187–206

    Article  Google Scholar 

  • Heitsch H, Römisch W (2006) Stability and scenario trees for multistage stochastic programs. In: Infanger G (ed) Stochastic Programming—The State of the Art (submitted)

  • Heitsch H, Römisch W (2007) A note on scenario reduction for two-stage stochastic programs. Oper Res Let 35: 731–736

    Article  Google Scholar 

  • Heitsch H, Römisch W (2008) Scenario tree modeling for multistage stochastic programs. Math Program (to appear)

  • Heitsch H, Römisch W, Strugarek C (2006) Stability of multistage stochastic programs. SIAM J Optim 17: 511–525

    Article  Google Scholar 

  • Henrion R, Küchler C, Römisch W (2007) Scenario reduction in stochastic programming with respect to discrepancy distances. Comp Optim Appl (to appear)

  • Henrion R, Küchler C, Römisch W (2008) Discrepancy distances and scenario reduction in two-stage stochastic integer programming. J Ind Manage Optim 4: 363–384

    Google Scholar 

  • Hochreiter R, Pflug GCh (2007) Financial scenario generation for stochastic multi-stage decision processes as facility location problems. Ann Oper Res 152: 257–272

    Article  Google Scholar 

  • Høyland K, Kaut M, Wallace SW (2003) A heuristic for moment-matching scenario generation. Comp Optim Appl 24: 169–185

    Article  Google Scholar 

  • Kuhn D (2005) Generalized bounds for convex multistage stochastic programs. Lecture Notes in Economics and Mathematical Systems, vol 548. Springer, Berlin

    Google Scholar 

  • Kuhn D (2008) Aggregation and discretization in multistage stochastic programming. Math Program 113: 61–94

    Article  Google Scholar 

  • Pennanen T (2009) Epi-convergent discretizations of multistage stochastic programs via integration quadratures. Math Program 116: 461–479

    Article  Google Scholar 

  • Rachev ST, Rüschendorf L (1998) Mass transportation problems, vol I, II. Springer, Berlin

    Google Scholar 

  • Römisch W (2003) Stability of stochastic programming problems. In: Ruszczyński A, Shapiro A (eds) Stochastic programming, Handbooks in Operations Research and Management Science, vol 10. Elsevier, Amsterdam, pp 483–554

    Google Scholar 

  • Römisch W, Wets RJ-B (2007) Stability of ε-approximate solutions to convex stochastic programs. SIAM J Optim 18: 961–979

    Article  Google Scholar 

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Correspondence to Werner Römisch.

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Heitsch, H., Römisch, W. Scenario tree reduction for multistage stochastic programs. Comput Manag Sci 6, 117–133 (2009).

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