Skip to main content

A warm-start approach for large-scale stochastic linear programs

Mathematical Programming volume 127, pages 371–397 (2011)Cite this article

Abstract

We describe a way of generating a warm-start point for interior point methods in the context of stochastic programming. Our approach exploits the structural information of the stochastic problem so that it can be seen as a structure-exploiting initial point generator. We solve a small-scale version of the problem corresponding to a reduced event tree and use the solution to generate an advanced starting point for the complete problem. The way we produce a reduced tree tries to capture the important information in the scenario space while keeping the dimension of the corresponding (reduced) deterministic equivalent small. We derive conditions which should be satisfied by the reduced tree to guarantee a successful warm-start of the complete problem. The implementation within the HOPDM and OOPS interior point solvers shows remarkable advantages.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price includes VAT (USA)
Tax calculation will be finalised during checkout.

References

  1. Benson H.Y., Shanno D.F.: An exact primal–dual penalty method approach to warmstarting interior-point methods for linear programming. Comput. Optim. Appl. 38, 371–399 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  2. Birge J.R.: Decomposition and partitioning methods for multistage stochastic linear programs. Oper. Res. 33, 989–1007 (1985)

    MathSciNet  MATH  Article  Google Scholar 

  3. Birge J.R., Dempster M.A.H., Gassmann H.I., Gunn E.A., King A.J., Wallace S.W.: A standard input format for multiperiod stochastic linear programs. Comm. Algorithm. Newsl. 17, 1–19 (1987)

    Google Scholar 

  4. Birge J.R., Louveaux F.: Introduction to Stochastic Programming. Springer Series in Operations Research, New York (1997)

    MATH  Google Scholar 

  5. Bixby R.E.: Solving real-world linear programs: a decade and more of progress. Oper. Res. 50, 3–15 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  6. Colombo M., Gondzio J.: Further development of multiple centrality correctors. Comput. Optim. Appl. 41, 277–305 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  7. Dupačová J., Gröwe-Kuska N., Römisch W.: Scenario reduction in stochastic programming. Math. Program. 95, 493–511 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  8. Gondzio J.: Multiple centrality corrections in a primal-dual method for linear programming. Comput. Optim. Appl. 6, 137–156 (2003)

    MathSciNet  Google Scholar 

  9. Gondzio J.: Warm start of the primal-dual method applied in the cutting-plane scheme. Math. Program. 83, 125–143 (1998)

    MathSciNet  MATH  Google Scholar 

  10. Gondzio V., Grothey A.: Reoptimization with the primal-dual interior point method. SIAM. J. Optim. 13, 842–864 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  11. Gondzio, J., Grothey, A.: A new unblocking technique to warmstart interior point methods based on sensitivity analysis. Technical Report MS-06-005, School of Mathematics, The University of Edinburgh, December 2006. Accepted for publication in SIAM Journal on Optimization

  12. Gondzio J., Grothey A.: Solving non-linear portfolio optimization problems with the primal-dual interior point method. Eur. J. Oper. Res. 181, 1012–1029 (2007)

    MathSciNet  Article  Google Scholar 

  13. Gondzio J., Sarkissian R.: Parallel interior-point solver for structured linear programs. Math. Program. 96, 561–584 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  14. Gondzio J., Vial J.-P.: Warm start and ɛ-subgradients in a cutting plane scheme for block-angular linear programs. Comput. Optim. Appl. 14, 17–36 (1999)

    MathSciNet  MATH  Article  Google Scholar 

  15. Hipolito A.L.: A weighted least squares study of robustness in interior point linear programming. Comput. Optim. Appl. 2, 29–46 (1993)

    MathSciNet  MATH  Article  Google Scholar 

  16. Høyland K., Kaut M., Wallace S.W.: A heuristic for moment-matching scenario generation. Comput. Optim. Appl. 24, 169–185 (2003)

    MathSciNet  Article  Google Scholar 

  17. John E., Yıldırım E.A.: Implementation of warm-start strategies in interior-point methods for linear programming in fixed dimensions. Comput. Optim. Appl. 41, 151–183 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  18. Kall P., Wallace S.W.: Stochastic Programming. Wiley, New York (1994)

    MATH  Google Scholar 

  19. Linderoth J., Wright S.J.: Decomposition algorithms for stochastic programming on a computational grid. Comput. Optim. Appl. 24, 207–250 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  20. Mehrotra S.: On the implementation of a primal-dual interior point method. SIAM. J. Optim. 2, 575–601 (1992)

    MathSciNet  MATH  Article  Google Scholar 

  21. Mitchell J.E.: Karmakar’s algorithm and combinatorial optimization problems. PhD thesis, Cornell University (1988)

  22. Mitchell J.E., Todd M.J.: Solving combinatorial optimization problems using Karmarkar’s algorithm. Math. Program. 56, 245–284 (1992)

    MathSciNet  MATH  Article  Google Scholar 

  23. Mulvey J.M., Ruszczyński A.: A new scenario decomposition method for large-scale stochastic optimization. Oper. Res. 43, 477–490 (1995)

    MathSciNet  MATH  Article  Google Scholar 

  24. Ouorou A.: Robust capacity assignment in telecommunications. Comput. Manag. Sci. 3, 285–305 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  25. Pflug G.C.: Scenario tree generation for multiperiod financial optimization by optimal discretization. Math. Program. 89, 251–271 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  26. Wright S.J.: Primal-dual Interior-point Methods. SIAM, Philadelphia (1997)

    MATH  Google Scholar 

  27. Yıldırım E.A., Wright S.J.: Warm-start strategies in interior-point methods for linear programming. SIAM J. Optim. 12, 782–810 (2002)

    MathSciNet  MATH  Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, Maxwell Institute, The University of Edinburgh, Mayfield Road, Edinburgh, EH9 3JZ, UK

    Marco Colombo, Jacek Gondzio & Andreas Grothey

Authors
  1. Marco Colombo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jacek Gondzio
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Andreas Grothey
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marco Colombo.

Additional information

This research has been supported by France Télécom.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Colombo, M., Gondzio, J. & Grothey, A. A warm-start approach for large-scale stochastic linear programs. Math. Program. 127, 371–397 (2011). https://doi.org/10.1007/s10107-009-0290-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-009-0290-9

Mathematics Subject Classification (2000)

  • 90C15
  • 90C51
  • 60C06