Quantified Linear Programs: A Computational Study

  • Thorsten Ederer
  • Ulf Lorenz
  • Alexander Martin
  • Jan Wolf
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6942)

Abstract

Quantified linear programs (QLPs) are linear programs with variables being either existentially or universally quantified. The integer variant (QIP) is PSPACE-complete, and the problem is similar to games like chess, where an existential and a universal player have to play a two-person-zero-sum game. At the same time, a QLP with n variables is a variant of a linear program living in \(\hbox{\rm I\kern - 0.15em R}^n\), and it has strong similarities with multi-stage stochastic linear programs with variable right-hand side. Our interest in QLPs stems from the fact that they are LP-relaxations of QIPs, which themselves are mighty modeling tools. In order to solve QLPs, we apply a nested decomposition algorithm. In a detailed computational study, we examine, how different structural properties like the number of universal variables, the number of universal variable blocks as well as their positions in the QLP influence the solution process.

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References

  1. 1.
    Achterberg, T., Koch, T., Martin, A.: Branching rules revisited. Operations Research Letters 33, 42–54 (2005)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Altenstedt, F.: Memory consumption versus computational time in nested benders decomposition for stochastic linear programming. Tech.Rep., Chalmers University, Goteborg (2003)Google Scholar
  3. 3.
    Bellmann, R.: Dynamic programming. Princeton University Press, Princeton (1957)MATHGoogle Scholar
  4. 4.
    Benders, J.F.: Partitioning procedures for solving mixed-variables programming problems. Numerische Mathematik 4(1), 238–252 (1962)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Birge, J., Donohue, C., Holmes, D., Svintsitski, O.: A parallel implementation of the nested decomposition algorithm for multistage stochastic linear programs. Math. Program. 75, 327–352 (1996)MathSciNetMATHGoogle Scholar
  6. 6.
    Birge, J., Louveaux, F.: A multicut algorithm for two-stage stochastic linear programs. European Journal of Operational Research 34(3), 384–392 (1988)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Birge, J., Louveaux, F.: Intro. to Stochastic Programming. Springer, Heidelberg (1997)MATHGoogle Scholar
  8. 8.
    Dempster, M., Thompson, R.: Parallelization and aggregation ofnested benders decomposition. Annals of Operations Research 81, 163–188 (1998), doi:10.1023/A:1018996821817MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Donninger, C., Kure, A., Lorenz, U.: Parallel brutus: The first distributed, fpga accelarated chess program. In: Proc. of 18th International Parallel & Distributed Processing Symposium (IPDPS). IEEE Computer Society, Santa Fe (2004)Google Scholar
  10. 10.
    Dyer, M., Stougie, L.: Computational complexity of stochastic programming problems. Math. Program. 106(3), 423–432 (2006)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Engell, S., Märkert, A., Sand, G., Schultz, R.: Aggregated scheduling of a multiproduct batch plant by two-stage stochastic integer programming. Optimiz. and Engineering 5 (2004)Google Scholar
  12. 12.
    Gassmann, H.: Mslip: A computer code for the multistage stochastic linear programming problem. Mathematical Programming 47(1-3), 407–423 (1990)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Ho, J., Sundarraj, R.P.: Distributed nested decomposition of staircase linear programs. ACM Trans. Math. Softw. 23, 148–173 (1997)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Kılınç Karzan, F., Nemhauser, G., Savelsbergh, M.: Information-based branching schemes for binary linear mixed integer problems. Mathematical Programming Computation (January 2010)Google Scholar
  15. 15.
    Kleywegt, A., Shapiro, A., Homem-De-Mello, T.: The sample average approximation method for stochastic discrete optimization. SIAM Jour. of Opt., 479–502 (2001)Google Scholar
  16. 16.
    König, F.G., Lübbecke, M., Möhring, R.H., Schäfer, G., Spenke, I.: Solutions to real-world instances of PSPACE-complete stacking. In: Arge, L., Hoffmann, M., Welzl, E. (eds.) ESA 2007. LNCS, vol. 4698, pp. 729–740. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  17. 17.
    Liebchen, C., Lübbecke, M., Möhring, R., Stiller, S.: The concept of recoverable robustness, linear programming recovery, and railway applications. Robust and online large-scale optimization, 1–27 (2009)Google Scholar
  18. 18.
    Lorenz, U., Martin, A., Wolf, J.: Polyhedral and algorithmic properties of quantified linear programs. In: de Berg, M., Meyer, U. (eds.) ESA 2010. LNCS, vol. 6346, pp. 512–523. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  19. 19.
    Megow, N., Vredeveld, T.: Approximation in preemptive stochastic online scheduling. In: Azar, Y., Erlebach, T. (eds.) ESA 2006. LNCS, vol. 4168, pp. 516–527. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  20. 20.
    Möhring, R., Schulz, A., Uetz, M.: Approximation in stochastic scheduling: The power of lp-based priority schedules. Journal of ACM 46(6), 924–942 (1999)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Ostrowski, J.P., Linderoth, J., Rossi, F., Smriglio, S.: Orbital branching. In: Fischetti, M., Williamson, D.P. (eds.) IPCO 2007. LNCS, vol. 4513, pp. 104–118. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  22. 22.
    Pochet, Y., Wolsey, L.: Production planning by mixed integer programming. Springer Series in Operations Research and Financial Engineering. Springer, New York (2006)MATHGoogle Scholar
  23. 23.
    Poojari, C., Beasley, J.: Improving benders decomposition using a genetic algorithm. European Journal of Operational Research 199(1), 89–97 (2009)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Rockafellar, R.T., Wets, R.-B.: Scenarios and policy aggregation in optimization under uncertainty. Math. Oper. Res. 16, 119–147 (1991)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Ruszczyński, A.: Parallel decomposition of multistage stochastic programming problems. Math. Program. 58, 201–228 (1993)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Ruszczyński, A.: Decomposition methods in stochastic programming. Math. Program. 79, 333–353 (1997)MathSciNetMATHGoogle Scholar
  27. 27.
    Schultz, R.: Stochastic programming with integer variables. Math. Progr. 97, 285–309 (2003)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Shmoys, D., Swamy, C.: Stochastic optimization is (almost) as easy as deterministic optimization. In: Proc. FOCS 2004, pp. 228–237 (2004)Google Scholar
  29. 29.
    Sirikum, J., Techanitisawad, A., Kachitvichyanukul, V.: A new efficient GA-benders’ decomposition method: For power generation expansion planning with emission controls. IEEE Transactions on Power Systems 22(3), 1092–1100 (2007)CrossRefGoogle Scholar
  30. 30.
    Subramani, K.: Analyzing selected quantified integer programs. In: Basin, D., Rusinowitch, M. (eds.) IJCAR 2004. LNCS (LNAI), vol. 3097, pp. 342–356. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  31. 31.
    Subramani, K.: On a decision procedure for quantified linear programs. Annals of Mathematics and Artificial Intelligence 51(1), 55–77 (2007)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Van Slyke, R.M., Wets, R.: L-Shaped linear programs with applications to optimal control and stochastic programming. SIAM Journal on Applied Mathematics 17(4), 638–663 (1969)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Verweij, B., Ahmed, S., Kleywegt, A.J., Nemhauser, G., Shapiro, A.: The sample average approximation method applied to stochastic routing problems: A computational study. Comput. Optim. Appl. 24, 289–333 (2003)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Zhang, L., Hermanns, H., Eisenbrand, F., Jansen, D.: Flow faster: Efficient decision algorithms for probabilistic simulations. Logical Methods in Computer Science 4(4) (2008)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Thorsten Ederer
    • 1
  • Ulf Lorenz
    • 1
  • Alexander Martin
    • 2
  • Jan Wolf
    • 1
  1. 1.Institute of MathematicsTechnische Universität DarmstadtGermany
  2. 2.Institute of MathematicsUniversität Erlangen-NürnbergGermany

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