On the Scenario-Tree Optimal-Value Error for Stochastic Programming Problems

Published Online:https://doi.org/10.1287/moor.2019.1043

References

  • [1] Bastin F, Cirillo C, Toint PL (2006) Convergence theory for nonconvex stochastic programming with an application to mixed logit. Math. Programming 108(2–3):207–234.CrossrefGoogle Scholar
  • [2] Bellman RE (1961) Adaptive Control Processes: A Guided Tour (Princeton University Press, Princeton, NJ).CrossrefGoogle Scholar
  • [3] Beraldi P, Bruni ME, Conforti D (2004) Designing robust emergency medical service via stochastic programming. Eur. J. Oper. Res. 158(1):183–193.CrossrefGoogle Scholar
  • [4] Bertocchi M, Consigli G, Dempster MA (2011) Stochastic Optimization Methods in Finance and Energy: New Financial Products and Energy Market Strategies (Springer, New York).CrossrefGoogle Scholar
  • [5] Birge JR, Louveaux F (2011) Introduction to Stochastic Programming, 2nd ed. (Springer, New York).CrossrefGoogle Scholar
  • [6] Colvin M, Maravelias CT (2008) A stochastic programming approach for clinical trial planning in new drug development. Comput. Chemical Engrg. 32(11):2626–2642.CrossrefGoogle Scholar
  • [7] Dick J, Pillichshammer F (2010) Digital Nets and Sequences: Discrepancy Theory and Quasi–Monte Carlo Integration (Cambridge University Press, Cambridge, UK).CrossrefGoogle Scholar
  • [8] Dupačová J, Gröwe-Kuska N, Römisch W (2003) Scenario reduction in stochastic programming. Math. Programming 95(3):493–511.CrossrefGoogle Scholar
  • [9] Dyer M, Stougie L (2006) Computational complexity of stochastic programming problems. Math. Programming 106(3):423–432.CrossrefGoogle Scholar
  • [10] Griebel M, Kuo FY, Sloan IH (2010) The smoothing effect of the ANOVA decomposition. J. Complexity 26(5):523–551.CrossrefGoogle Scholar
  • [11] Griebel M, Kuo FY, Sloan IH (2013) The smoothing effect of integration in ℛd and the ANOVA decomposition. Math. Comput. 82(281):383–400.CrossrefGoogle Scholar
  • [12] Growe-Kuska N, Heitsch H, Romisch W (2003) Scenario reduction and scenario tree construction for power management problems. Borghetti A, Nucci CA, Paolone M, eds. 2003 IEEE Bologna Power Tech Conf. Proc., Bologna, Italy, vol. 3 (IEEE, New York), 1–7.Google Scholar
  • [13] Hanasusanto GA, Kuhn D, Wiesemann W (2016) A comment on “Computational complexity of stochastic programming problems.” Math. Programming 159(1):557–569.CrossrefGoogle Scholar
  • [14] Heitsch H, Leövey H, Römisch W (2016) Are Quasi-Monte Carlo algorithms efficient for two-stage stochastic programs? Comput. Optim. Appl. 65(3):567–603.CrossrefGoogle Scholar
  • [15] Heitsch H, Römisch W (2003) Scenario reduction algorithms in stochastic programming. Comput. Optim. Appl. 24(2–3):187–206.CrossrefGoogle Scholar
  • [16] Heitsch H, Römisch W (2009) Scenario tree modeling for multistage stochastic programs. Math. Programming 118(2):371–406.CrossrefGoogle Scholar
  • [17] Heitsch H, Römisch W, Strugarek C (2006) Stability of multistage stochastic programs. SIAM J. Optim. 17(2):511–525.CrossrefGoogle Scholar
  • [18] Hickernell F, Sloan IH, Wasilkowski GW (2004) On tractability of weighted integration over bounded and unbounded regions in ℛs. Math. Comput. 73(248):1885–1901.CrossrefGoogle Scholar
  • [19] Holtz M (2010) Sparse Grid Quadrature in High Dimensions with Applications in Finance and Insurance (Springer, New York).Google Scholar
  • [20] Homem-de Mello T (2008) On rates of convergence for stochastic optimization problems under non-independent and identically distributed sampling. SIAM J. Optim. 19(2):524–551.CrossrefGoogle Scholar
  • [21] Høyland K, Kaut M, Wallace SW (2003) A heuristic for moment-matching scenario generation. Comput. Optim. Appl. 24(2–3):169–185.CrossrefGoogle Scholar
  • [22] Høyland K, Wallace SW (2001) Generating scenario trees for multistage decision problems. Management Sci. 47(2):295–307.LinkGoogle Scholar
  • [23] Keutchayan J, Gendreau M, Bastin F (2018) Problem-driven scenario trees in multistage stochastic optimization: An illustration in option pricing. Working paper, Polytechnique Montréal, Montreal, Canada.Google Scholar
  • [24] Keutchayan J, Gendreau M, Saucier A (2017) Quality evaluation of scenario-tree generation methods for solving stochastic programming problems. Comput. Management Sci. 14(3):333–365.CrossrefGoogle Scholar
  • [25] Keutchayan J, Munger D, Gendreau M, Bastin F (2018) The figure of demerit: A quality measure for the discretization of probability distributions in multistage stochastic optimization. Working paper, Polytechnique Montréal, Montreal, Canada.Google Scholar
  • [26] King AJ, Wallace SW (2012) Modeling with Stochastic Programming (Springer, New York).CrossrefGoogle Scholar
  • [27] Koivu M (2005) Variance reduction in sample approximations of stochastic programs. Math. Programming 103(3):463–485.CrossrefGoogle Scholar
  • [28] Kovacevic RM, Pflug GC, Vespucci MT (2013) Handbook of Risk Management in Energy Production and Trading (Springer, New York).CrossrefGoogle Scholar
  • [29] Kuo FY, Schwab C, Sloan IH (2011) Quasi-Monte Carlo methods for high-dimensional integration: The standard (weighted Hilbert space) setting and beyond. ANZIAM J. 53(1):1–37.CrossrefGoogle Scholar
  • [30] Lemieux C (2009) Monte Carlo and Quasi-Monte Carlo Sampling (Springer, New York).Google Scholar
  • [31] Leövey H, Römisch W (2015) Quasi-Monte Carlo methods for linear two-stage stochastic programming problems. Math. Programming 151(1):315–345.CrossrefGoogle Scholar
  • [32] Mak WK, Morton DP, Wood KR (1999) Monte Carlo bounding techniques for determining solution quality in stochastic programs. Oper. Res. Lett. 24(1):47–56.CrossrefGoogle Scholar
  • [33] Markowitz H (1952) Portfolio selection. J. Finance 7(1):77–91.Google Scholar
  • [34] Novak E, Ullrich M, Woźniakowski H, Zhang S (2018) Reproducing kernels of Sobolev spaces on ℛd and applications to embedding constants and tractability. Anal. Appl. 16(5):693–715.Google Scholar
  • [35] Novak E, Woźniakowski H (2010) Tractability of multivariate problems - Volume II: Standard Information for Functionals (European Mathematical Society, Zürich).Google Scholar
  • [36] Olsen P (1976) Discretizations of multistage stochastic programming problems. Wets RJB, ed. Stochastic Systems: Modeling, Identification and Optimization, vol. 2 (Springer, New York), 111–124.CrossrefGoogle Scholar
  • [37] Pagès G, Pham H, Printems J (2004) Optimal quantization methods and applications to numerical problems in finance. Rachev ST, ed. Handbook of Computational and Numerical Methods in Finance (Birkhäuser, Boston), 253–297.CrossrefGoogle Scholar
  • [38] Pennanen T (2009) Epi-convergent discretizations of multistage stochastic programs via integration quadratures. Math. Programming 116(1–2):461–479.CrossrefGoogle Scholar
  • [39] Pennanen T, Koivu M (2005) Epi-convergent discretizations of stochastic programs via integration quadratures. Numerische Mathematik 100(1):141–163.CrossrefGoogle Scholar
  • [40] Pflug GC (2001) Scenario tree generation for multiperiod financial optimization by optimal discretization. Math. Programming 89(2):251–271.CrossrefGoogle Scholar
  • [41] Pflug GC, Pichler A (2011) Approximations for probability distributions and stochastic optimization problems. Bertocchi M, Consigli G, Dempster MAH, eds. Stochastic Optimization Methods in Finance and Energy (Springer, New York), 343–387.CrossrefGoogle Scholar
  • [42] Pflug GC, Pichler A (2012) A distance for multistage stochastic optimization models. SIAM J. Optim. 22(1):1–23.CrossrefGoogle Scholar
  • [43] Pflug GC, Pichler A (2015) Dynamic generation of scenario trees. Comput. Optim. Appl. 62(3):641–668.CrossrefGoogle Scholar
  • [44] Pflug GC, Pichler A (2016) Multistage Stochastic Optimization (Springer, New York).Google Scholar
  • [45] Powell WB, Topaloglu H (2003) Stochastic programming in transportation and logistics. Ruszczyński A, Shapiro A, eds. Handbooks in Operations Research and Management Science: Stochastic Programming, vol. 10 (Elsevier, Amsterdam), 555–635.Google Scholar
  • [46] Prékopa A (1995) Stochastic Programming (Kluwer, Dordrecht, Netherlands).CrossrefGoogle Scholar
  • [47] Rockafellar RT, Wets RJB (2009) Variational Analysis (Springer, New York).Google Scholar
  • [48] Römisch W (2003) Stability of stochastic programming problems. Ruszczyński A, Shapiro A, eds. Handbooks in Operations Research and Management Science: Stochastic Programming, vol. 10 (Elsevier, Amsterdam), 483–554.Google Scholar
  • [49] Ruszczyński A (2003) Decomposition methods. Ruszczyński A, Shapiro A, eds. Handbooks in Operations Research and Management Science: Stochastic Programming, vol. 10 (Elsevier, Amsterdam), 141–211.Google Scholar
  • [50] Ruszczyński A, Shapiro A, eds. (2003) Handbooks in Operations Research and Management Science: Stochastic Programming, vol. 10 (Elsevier, Amsterdam).Google Scholar
  • [51] Ruszczyński A, Shapiro A (2003) Stochastic programming models. Ruszczyński A, Shapiro A, eds. Handbooks in Operations Research and Management Science: Stochastic Programming, vol. 10 (Elsevier, Amsterdam), 1–64.Google Scholar
  • [52] Shapiro A (2003) Monte Carlo sampling methods. Ruszczyński A, Shapiro A, eds. Handbooks in Operations Research and Management Science: Stochastic Programming, vol. 10 (Elsevier, Amsterdam), 353–425.Google Scholar
  • [53] Shapiro A, Homem-de Mello T (1998) A simulation-based approach to two-stage stochastic programming with recourse. Math. Programming 81(3):301–325.CrossrefGoogle Scholar
  • [54] Shapiro A, Homem-de Mello T (2000) On the rate of convergence of optimal solutions of Monte Carlo approximations of stochastic programs. SIAM J. Optim. 11(1):70–86.CrossrefGoogle Scholar
  • [55] Sloan IH, Woźniakowski H (1998) When are quasi-Monte Carlo algorithms efficient for high dimensional integrals? J. Complexity 14(1):1–33.CrossrefGoogle Scholar
  • [56] Villani C (2008) Optimal Transport: Old and New (Springer, New York).Google Scholar
  • [57] Yen JW, Birge JR (2006) A stochastic programming approach to the airline crew scheduling problem. Transportation Sci. 40(1):3–14.LinkGoogle Scholar
  • [58] Yu LY, Ji XD, Wang SY (2003) Stochastic programming models in financial optimization: A survey. Advanced Modeling Optim. 5(1):1–26.Google Scholar
  • [59] Ziemba WT (2003) The Stochastic Programming Approach to Asset, Liability, and Wealth Management (AIMR, Charlottesville, VA).Google Scholar
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