A Scalable Parallel Interior Point Algorithm for Stochastic Linear Programming and Robust Optimization

  • Dafeng Yang
  • Stavros A. Zenios
Chapter

Abstract

We present a computationally efficient implementation of an interior point algorithm for solving large-scale problems arising in stochastic linear programming and robust optimization. A matrix factorization procedure is employed that exploits the structure of the constraint matrix, and it is implemented on parallel computers. The implementation is perfectly scalable. Extensive computational results are reported for a library of standard test problems from stochastic linear programming, and also for robust optimization formulations. The results show that the codes are efficient and stable for problems with thousands of scenarios. Test problems with 130 thousand scenarios, and a deterministic equivalent linear programming formulation with 2.6 million constraints and 18.2 million variables, are solved successfully.

Keywords

planning under uncertainty parallel computing optimization software 
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References

  1. 1.
    A.J. Berger, J.M. Mulvey, and A. Ruszczyński, “An extension of the DQA algorithm to convex stochastic programs,” SIAM Journal on Optimization, vol. 4, no. 4, pp. 735–753, 1994.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    J.R. Birge and L. Qi, “Computing block-angular Karmarkar projections with applications to stochastic programming,” Management Science, vol. 34, pp. 1472–1479, Dec. 1988.Google Scholar
  3. 3.
    J.R. Birge and D.F. Holmes, “Efficient solution of two-stage stochastic linear programs using interior point methods,” Computational Optimization and Applications, vol. 1, pp. 245–276, 1992.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Y. Censor and S.A. Zenios, Parallel Optimization: Theory, Algorithms and Applications, Oxford University Press: Oxford, England, 1997 (in print).MATHGoogle Scholar
  5. 5.
    G.B. Dantzig, “Planning under uncertainty using parallel computing,” Annals of Operations Research, vol. 14, pp. 1–16, 1988.CrossRefMathSciNetGoogle Scholar
  6. 6.
    G.B. Dantzig, J.K. Ho, and G. Infanger, “Solving stochastic linear programs on a hypercube multicomputer,” Technical report sol 91-10, Operations Research Department, Stanford University, Stanford, CA, 1991.Google Scholar
  7. 7.
    E.R. Jessup, D. Yang, and S.A. Zenios, “Parallel factorization of structured matrices arising in stochastic programming, SIAM Journal on Optimization, vol. 4, no. 4, pp. 833–846, 1994.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    P. Kall and S.W. Wallace, Stochastic Programming, John Wiley & Sons: New York, 1994.MATHGoogle Scholar
  9. 9.
    C.E. Leiserson, Z.S. Abuhamdeh, D.C. Douglas, C.R. Feynman, M.N. Ganmukhi, J.V. Hill, W.D. Hillis, B.C. Kuszmaul, M.A. St. Pierre, D.S. Wells, M.C. Wong, S.-W. Yang, and R. Zak, “The network architecture of the Connection Machine CM-5,” Manuscript, Thinking Machines Corporation, Cambridge, Massachusetts 02142, 1992.Google Scholar
  10. 10.
    R.D.C. Monteiro and I. Adler, “Interior path-following primal-dual algorithms. Part II: Convex quadratic programming,” Mathematical Programming, vol. 44, pp. 43–66, 1989.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    J.M. Mulvey and A. Ruszczyński, “A new scenario decomposition method for large-scale stochastic optimization,” Operations Research, vol. 43, pp. 477–490, 1994.Google Scholar
  12. 12.
    J.M. Mulvey, R.J. Vanderbei, and S.A. Zenios, “Robust optimization of large scale systems,” Operations Research, vol. 43, pp. 264–281, 1995.MATHMathSciNetGoogle Scholar
  13. 13.
    J.M. Mulvey and H. Vladimireu, “Evaluation of a parallel hedging algorithm for stochastic network programming,” in Impacts of Recent Computer Advances on Operations Research, R. Sharda, B.L. Golden, E. Wasil, O. Balci, and W. Stewart (Eds.), North-Holland, New York, USA, 1989.Google Scholar
  14. 14.
    E. Ng and B. Peyton, “A supernodal Cholesky factorization algorithm for shared-memory multiprocessors,” SIAM Journal on Scientific and Statistical Computing, vol. 14, pp. 761–769, 1993.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    S.S. Nielsen and S.A. Zenios, “A massively parallel algorithm for nonlinear stochastic network problems,” Operations Research, vol. 41, no. 2, pp. 319–337, 1993.MATHMathSciNetGoogle Scholar
  16. 16.
    S.S. Nielsen and S.A. Zenios, “Scalable parallel Benders decomposition for stochastic linear programming,” Technical report, Management Science and Information Systems Department, University of Texas at Austin, Austin, TX, 1994.Google Scholar
  17. 17.
    S. Sen, R.D. Doverspike, and S. Cosares, “Network planning with random demand,” Working paper, Systems and Industrial Engineering Department, University of Arizona, Tucson, AZ, 1992.Google Scholar
  18. 18.
    R.J. Vanderbei, “LOQO user’s manual,” Technical report SOR 92-5, Department of Civil Engineering and Operations Research, Princeton University, Princeton, NJ, 1992.Google Scholar
  19. 19.
    R.J. Vanderbei and T.J. Carpenter, “Symmetric indefinite systems for interior point methods,” Mathematical Programming, vol. 58, pp. 1–32, 1993.MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    R.J.-B. Wets, “Stochastic programming,” in Handbooks in Operations Research and Management Science, G.L. Nemhauser, A.H.G. Rinnooy Kan, and M.J. Todd (Eds.), vol. 1, pp. 573–629, North-Holland, Amsterdam, 1989.Google Scholar

Copyright information

© Kluwer Academic Publishers 1997

Authors and Affiliations

  • Dafeng Yang
    • 1
  • Stavros A. Zenios
    • 2
  1. 1.Operations and Information Management, The Wharton SchoolUniversity of PennsylvaniaPhiladelphia
  2. 2.Department of Public and Business AdministrationUniversity of CyprusNicosiaCyprus

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