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MIDAS: A mixed integer dynamic approximation scheme

  • A. B. PhilpottEmail author
  • F. Wahid
  • J. F. Bonnans
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Abstract

Mixed integer dynamic approximation scheme (MIDAS) is a new sampling-based algorithm for solving finite-horizon stochastic dynamic programs with monotonic Bellman functions. MIDAS approximates these value functions using step functions, leading to stage problems that are mixed integer programs. We provide a general description of MIDAS, and prove its almost-sure convergence to a \(2T\varepsilon \)-optimal policy for problems with T stages when the Bellman functions are known to be monotonic, and the sampling process satisfies standard assumptions.

Keywords

Stochastic programming Approximate dynamic programming Sampling Mixed-integer programming 

Mathematics Subject Classification

90C15 90C39 

Notes

Funding

Funding was provided by PGMO, EDF, Meridian Energy Limited and the New Zealand Marsden Fund.

References

  1. 1.
    Abgottspon, H., Njalsson, K., Bucher, M.A., Andersson, G.: Risk-averse medium-term hydro optimization considering provision of spinning reserves. In: 2014 International Conference on Probabilistic Methods Applied to Power Systems (PMAPS), pp. 1–6. IEEE (2014)Google Scholar
  2. 2.
    Bertsekas, D.P.: Dynamic Programming and Optimal Control, vol. 1. Athena Scientific, Belmont (2005)zbMATHGoogle Scholar
  3. 3.
    Birge, J.R.: Decomposition and partitioning methods for multistage stochastic linear programs. Oper. Res. 33(5), 989–1007 (1985)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bonnans, J.F., Cen, Z., Christel, Th: Energy contracts management by stochastic programming techniques. Ann. Oper. Res. 200, 199–222 (2012)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Cerisola, S., Latorre, J.M., Ramos, A.: Stochastic dual dynamic programming applied to nonconvex hydrothermal models. Eur. J. Oper. Res. 218(3), 687–697 (2012)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Chen, Z., Powell, W.B.: Convergent cutting-plane and partial-sampling algorithm for multistage stochastic linear programs with recourse. J. Optim. Theory Appl. 102(3), 497–524 (1999)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Donohue, C.J., Birge, J.R.: The abridged nested decomposition method for multistage stochastic linear programs with relatively complete recourse. Algorithmic Oper. Res. 1(1), 20–30 (2006)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Downward, A., Dowson, O., Baucke, R.: Stochastic dual dynamic programming with stagewise dependent objective uncertainty. Available on Optimization Online. http://www.optimization-online.org/DB_FILE/2018/02/6454.pdf (2018)
  9. 9.
    Girardeau, P., Leclere, V., Philpott, A.B.: On the convergence of decomposition methods for multistage stochastic convex programs. Math. Oper. Res. 40(1), 130–145 (2014)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Gjelsvik, A., Mo, B., Haugstad, A.: Long-and medium-term operations planning and stochastic modelling in hydro-dominated power systems based on stochastic dual dynamic programming. In: Handbook of Power Systems I, pp. 33–55. Springer (2010)Google Scholar
  11. 11.
    Guigues, V.: Convergence analysis of sampling-based decomposition methods for risk-averse multistage stochastic convex programs. SIAM J. Optim. 26(4), 2468–2494 (2016)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Guigues, V.: Inexact cuts in deterministic and stochastic dual dynamic programming applied to linear optimization problems. arXiv:1801.04243 (2018)
  13. 13.
    Hindsberger, M., Philpott, A.B.: Resa: a method for solving multistage stochastic linear programs. J. Appl. Oper. Res. 6(1), 2–15 (2014)Google Scholar
  14. 14.
    Hjelmeland, M.M., Zou, J., Helseth, A., Ahmed, S.: Nonconvex medium-term hydropower scheduling by stochastic dual dynamic integer programming. IEEE Trans. Sustain. Energy 10(1), 481–490 (2019)Google Scholar
  15. 15.
    Pereira, M.V.F., Pinto, L.M.V.G.: Multistage stochastic optimization applied to energy planning. Math. Program. 52(1–3), 359–375 (1991)zbMATHGoogle Scholar
  16. 16.
    Philpott, A.B., Guan, Z.: On the convergence of stochastic dual dynamic programming and related methods. Oper. Res. Lett. 36(4), 450–455 (2008)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Powell, W.B.: Approximate Dynamic Programming: Solving the Curses of Dimensionality, vol. 703. Wiley, Hoboken (2007)zbMATHGoogle Scholar
  18. 18.
    Shapiro, A.: On complexity of multistage stochastic programs. Oper. Res. Lett. 34(1), 1–8 (2006)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Shapiro, A.: Analysis of stochastic dual dynamic programming method. Eur. J. Oper. Res. 209(1), 63–72 (2011)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Steeger, G., Rebennack, S.: Strategic bidding for multiple price-maker hydroelectric producers. IIE Trans. 47(9), 1013–1031 (2015)Google Scholar
  21. 21.
    Thome, F., Pereira, M.V.F., Granville, S., Fampa, M.: Non-convexities representation on hydrothermal operation planning using SDDP. Technical report, working paper. https://www.psr-inc.com/publications/scientific-production/papers/?current=t606 (2013). Accessed 6 Feb 2019
  22. 22.
    Wahid, F.: River optimization: short-term hydro bidding under uncertainty. Ph.D. thesis, University of Auckland/Ecole Polytechnique (2017)Google Scholar
  23. 23.
    Zou, J., Ahmed, S., Sun, X.A.: Stochastic dual dynamic integer programming. Math. Program.  https://doi.org/10.1007/s10107-018-1249-5 (2018)

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2019

Authors and Affiliations

  1. 1.Electric Power Optimization CentreUniversity of AucklandAucklandNew Zealand
  2. 2.Artelys ConsultingParisFrance
  3. 3.INRIA, Ecole PolytechniqueSaclayFrance

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