Bounding Multistage Stochastic Programs: A Scenario Tree Based Approach

  • Francesca Maggioni
  • Elisabetta Allevi
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 217)


Multistage mixed-integer stochastic programs are among the most challenging optimization problems combining stochastic programs and discrete optimization problems. Approximation techniques which provide lower and upper bounds to the optimal value are very useful in practice. In this paper we present a critic summary of the results in Maggioni et al., J Optim Theory Appl 163:200–229 (2014), [4] and in Maggioni et al., Comput Manag Sci 13:423–457 (2016), [5] where we consider bounds based on the assumption that a sufficiently large discretized scenario tree describing the problem uncertainty is given but is unsolvable. Bounds based on group subproblems, quality of the deterministic solution and rolling-horizon approximation will be then discussed and compared.


Multistage stochastic programs Bounds Group subproblems 


  1. 1.
    Crainic, T.G., Maggioni, F., Perboli, G., Rei, W.: Reduced Cost-Based Variable Fixing in Two-Stage Stochastic Programming (2017) Available via CIRRELT ReportsGoogle Scholar
  2. 2.
    Escudero, L.F., Garín, A., Merino, M., Pérez, G.: The value of the stochastic solution in multistage problems. Top 15, 48–64 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Madansky, A.: Inequalities for stochastic linear programming problems. Manag. Sci. 6, 197–204 (1960)Google Scholar
  4. 4.
    Maggioni, F., Allevi, E., Bertocchi, M.: Bounds in multistage linear stochastic programming. J. Optim. Theory Appl. 163(1), 200–229 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Maggioni, F., Allevi, E., Bertocchi, M.: Monotonic bounds in multistage mixed-integer linear stochastic programming. Comput. Manag. Sci. 13(3), 423–457 (2016)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Maggioni, F., Kaut, M., Bertazzi, L.: Stochastic Optimization models for a single-sink transportation problem. Comput. Manag. Sci 6, 251–267 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Maggioni, F., Pflug, G.C.: Bounds and approximations for multistage stochastic programs. SIAM J. Optim. 26(1), 831–855 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Maggioni, F., Pflug, G.C.: Guaranteed Bounds for General Non-discrete Multistage Risk-Averse Stochastic Optimization Programs (2017) Submitted to SIOPTGoogle Scholar
  9. 9.
    Maggioni, F., Wallace, W.S.: Analyzing the quality of the expected value solution in stochastic programming. Ann. Oper. Res. 200(1), 37–54 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Sandikçi, B., Kong, N., Schaefer, A.J.: A hierarchy of bounds for stochastic mixed-integer programs. Math. Program. Ser. A 138(1), 253–272 (2012)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Shapiro, A., Dencheva, D., Ruszczyński, A.: Lectures on Stochastic Programming: Modeling and Theory. MPS-SIAM Series on Optimization (2009)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.University Of BergamoBergamoItaly
  2. 2.University Of BresciaBresciaItaly

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