Skip to main content
SpringerLink
Log in

An Integer Programming and Decomposition Approach to General Chance-Constrained Mathematical Programs

  • Conference paper
Integer Programming and Combinatorial Optimization (IPCO 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6080))

Abstract

We present a new approach for exactly solving general chance constrained mathematical programs having discrete distributions. Such problems have been notoriously difficult to solve due to nonconvexity of the feasible region, and currently available methods are only able to find provably good solutions in certain very special cases. Our approach uses both decomposition, to enable processing subproblems corresponding to one possible outcome at a time, and integer programming techniques, to combine the results of these subproblems to yield strong valid inequalities. Computational results on a chance-constrained two-stage problem arising in call center staffing indicate the approach works significantly better than both an existing mixed-integer programming formulation and a simple decomposition approach that does not use strong valid inequalities. Thus, the strength of this approach results from the successful merger of stochastic programming decomposition techniques with integer programming techniques for finding strong valid inequalities.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Luedtke, J., Ahmed, S.: A sample approximation approach for optimization with probabilistic constraints. SIAM J. Optim. 19, 674–699 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  2. Charnes, A., Cooper, W.W., Symonds, G.H.: Cost horizons and certainty equivalents: an approach to stochastic programming of heating oil. Manage. Sci. 4, 235–263 (1958)

    Article  Google Scholar 

  3. Prékopa, A.: On probabilistic constrained programmming. In: Kuhn, H.W. (ed.) Proceedings of the Princeton Symposium on Mathematical Programming, Princeton, NJ, pp. 113–138. Princeton University Press, Princeton (1970)

    Google Scholar 

  4. Charnes, A., Cooper, W.W.: Deterministic equivalents for optimizing and satisficing under chance constraints. Oper. Res. 11, 18–39 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  5. Calafiore, G., El Ghaoui, L.: On distributionally robust chance-constrained linear programs. J. Optim. Theory Appl. 130, 1–22 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  6. Beraldi, P., Ruszczyński, A.: The probabilistic set-covering problem. Oper. Res. 50, 956–967 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  7. Dentcheva, D., Prékopa, A., Ruszczyński, A.: Concavity and efficient points of discrete distributions in probabilistic programming. Math. Program. 89, 55–77 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  8. Luedtke, J., Ahmed, S., Nemhauser, G.L.: An integer programming approach for linear programs with probabilistic constraints. Math. Program. 12, 247–272 (2010)

    Article  MathSciNet  Google Scholar 

  9. Saxena, A., Goyal, V., Lejeune, M.: MIP reformulations of the probabilistic set covering problem. Math. Program. 121, 1–31 (2009)

    Article  MathSciNet  Google Scholar 

  10. Ruszczyński, A.: Probabilistic programming with discrete distributions and precedence constrained knapsack polyhedra. Math. Program. 93, 195–215 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  11. Tanner, M., Ntaimo, L.: IIS branch-and-cut for joint chance-constrained programs with random technology matrices (2008)

    Google Scholar 

  12. Ben-Tal, A., Nemirovski, A.: Robust solutions of linear programming problems contaminated with uncertain data. Math. Program. 88, 411–424 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  13. Bertsimas, D., Sim, M.: The price of robustness. Oper. Res. 52, 35–53 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  14. Calafiore, G., Campi, M.: Uncertain convex programs: randomized solutions and confidence levels. Math. Program. 102, 25–46 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  15. Nemirovski, A., Shapiro, A.: Scenario approximation of chance constraints. In: Calafiore, G., Dabbene, F. (eds.) Probabilistic and Randomized Methods for Design Under Uncertainty, pp. 3–48. Springer, London (2005)

    Google Scholar 

  16. Nemirovski, A., Shapiro, A.: Convex approximations of chance constrained programs. SIAM J. Optim. 17, 969–996 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  17. Erdoğan, E., Iyengar, G.: Ambiguous chance constrained problems and robust optimization. Math. Program. 107, 37–61 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  18. Erdoğan, E., Iyengar, G.: On two-stage convex chance constrained problems. Math. Meth. Oper. Res. 65, 115–140 (2007)

    Article  MATH  Google Scholar 

  19. Luedtke, J., Ahmed, S., Nemhauser, G.: An integer programming approach for linear programs with probabilistic constraints. In: Fischetti, M., Williamson, D.P. (eds.) IPCO 2007. LNCS, vol. 4513, pp. 410–423. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  20. Birge, J., Louveaux, F.: Introduction to stochastic programming. Springer, New York (1997)

    MATH  Google Scholar 

  21. Van Slyke, R., Wets, R.J.: L-shaped linear programs with applications to optimal control and stochastic programming. SIAM J. Appl. Math. 17, 638–663 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  22. Higle, J.L., Sen, S.: Stochastic decomposition: an algorithm for two-stage stochastic linear programs. Math. Oper. Res. 16, 650–669 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  23. Shen, S., Smith, J., Ahmed, S.: Expectation and chance-constrained models and algorithms for insuring critical paths (2009) (submitted for publication)

    Google Scholar 

  24. Codato, G., Fischetti, M.: Combinatorial benders’ cuts for mixed-integer linear programming. Oper. Res. 54, 756–766 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  25. Atamtürk, A., Nemhauser, G.L., Savelsbergh, M.W.P.: The mixed vertex packing problem. Math. Program. 89, 35–53 (2000)

    MATH  MathSciNet  Google Scholar 

  26. Günlük, O., Pochet, Y.: Mixing mixed-integer inequalities. Math. Program. 90, 429–457 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  27. Linderoth, J., Savelsbergh, M.: A computational study of search strategies for mixed integer programming. INFORMS J. Comput. 11, 173–187 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  28. Applegate, D., Bixby, R., Chvátal, V., Cook, W.: Finding cuts in the TSP. Technical Report 95-05, DIMACS (1995)

    Google Scholar 

  29. Achterberg, T., Koch, T., Martin, A.: Branching rules revisited. Oper. Res. Lett. 33, 42–54 (2004)

    Article  MathSciNet  Google Scholar 

  30. Gurvich, I., Luedtke, J., Tezcan, T.: Call center staffing with uncertain arrival rates: a chance-constrained optimization approach. Technical report (2009)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Industrial and Systems Engineering, University of Wisconsin-Madison, Madison, WI, 53706, USA

    James Luedtke

Authors
  1. James Luedtke
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institute of Mathematics, École Polytechnique Féderale de Lausanne, 1015, Lausanne, Switzerland

    Friedrich Eisenbrand

  2. McGill University, 805 Sherbrooke West, H3A 2K6, Montreal, Quebec, Canada

    F. Bruce Shepherd

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Luedtke, J. (2010). An Integer Programming and Decomposition Approach to General Chance-Constrained Mathematical Programs. In: Eisenbrand, F., Shepherd, F.B. (eds) Integer Programming and Combinatorial Optimization. IPCO 2010. Lecture Notes in Computer Science, vol 6080. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13036-6_21

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-13036-6_21

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-13035-9

  • Online ISBN: 978-3-642-13036-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation