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Sensitivity method for basis inverse representation in multistage stochastic linear programming problems


A version of the simplex method for solving stochastic linear control problems is presented. The method uses a compact basis inverse representation that extensively exploits the original problem data and takes advantage of the supersparse structure of the problem. Computational experience indicates that the method is capable of solving large problems.

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  1. 1.

    Murtagh, B.,Advanced Linear Programming Computation and Practice, McGraw-Hill, New York, New York, 1981.

  2. 2.

    Bisschop, J., andMeeraus, A.,Matrix Augmentation and Structure Preservation in Linearly Constrained Control Problems, Mathematical Programming, Vol. 18, pp. 7–15, 1980.

  3. 3.

    Fourer, R.,Solving Staircase Linear Programs by the Simplex Method, Part 1: Inversion, Mathematical Programming, Vol. 23, pp. 274–313, 1982.

  4. 4.

    Fourer, R.,Solving Staircase Linear Programs by the Simplex Method, Part 2: Pricing, Mathematical Programming, Vol. 25, pp. 251–292, 1983.

  5. 5.

    Perold, A. F., andDantzig, G. B.,A Basis Factorization Method for Block Triangular Linear Programs, Sparse Matrix Proceedings 1978, Edited by I. S. Duff and G. W. Stewart, SIAM, Philadelphia, Pennsylvania, pp. 283–312, 1979.

  6. 6.

    Wets, R. J. B.,Large-Scale Linear Programming Techniques, Numerical Techniques in Stochastic Optimization, Edited by Yu. Ermoliev and R. J. B. Wets, Springer-Verlag, Berlin, Germany, 1988.

  7. 7.

    Ho, J. K., andManne, A. S.,Nested Decomposition for Dynamic Energy Models, Mathematical Programming, Vol. 6, pp. 121–140, 1974.

  8. 8.

    Ho, J. K., andLoute, E.,A Comparative Study of Two Methods for Staircase Linear Programs, ACM Transactions on Mathematical Software, Vol. 5, pp. 17–30, 1979.

  9. 9.

    Wittrock, R.,Dual Nested Decomposition of Staircase Linear Programs, Mathematical Programming Study, Vol. 24, pp. 65–86, 1985.

  10. 10.

    Ruszcyński, A.,A Regularized Decomposition Method for Minimizing a Sum of Polyhedral Functions, Mathematical Programming, Vol. 35, pp. 309–333, 1986.

  11. 11.

    Ruszczyński, A.,Modern Techniques for Linear Dynamic and Stochastic Programs, Aspiration-Based Decision Support Systems, Edited by A. Lewandowski and A. Wierzbicki, Springer-Verlag, Berlin, Germany, pp. 48–67, 1989.

  12. 12.

    Birge, J. R.,Decomposition and Partitioning Methods for Multistage Stochastic Linear Programs, Operations Research, Vol. 33, pp. 989–1007, 1985.

  13. 13.

    Rockafellar, R. T., andWets, R. J. B.,Scenarios and Policy Aggregation in Optimization under Uncertainty, Mathematics of Operations Research, Vol. 16, pp. 119–147, 1991.

  14. 14.

    Ruszczyński, A.,Parallel Decomposition of Multistage Stochastic Programming Problems, Working Paper WP-88-094, IIASA, Laxenburg, Austria, 1988.

  15. 15.

    Bartels, R. H., andGolub, G. H.,The Simplex Method of Linear Programming Using LU Decomposition, Communications of the ACM, Vol. 12, pp. 266–268, 1969.

  16. 16.

    Forrest, J. J. H., andTomlin, J. A.,Updating Triangular Factors of the Basis to Maintain Sparsity in the Product Form Simplex Method, Mathematical Programming, Vol. 2, pp. 263–279, 1972.

  17. 17.

    Saunders, M.,A Fast, Stable Implementation of the Simplex Method Using Bartels-Golub Updating, Sparse Matrix Computations, Edited by J. R. Bunch and D. J. Rose, Academic Press, New York, New York, pp. 213–226, 1976.

  18. 18.

    Reid, J.,A Sparsity-Exploiting Variant of the Bartels-Golub Decomposition for Linear Programming Bases, Mathematical Programming, Vol. 24, pp. 55–69, 1982.

  19. 19.

    Murtagh, B., andSaunders, M.,Minos 5.0. User's Guide, Technical Report SOL-83-20, Department of Operations Research, Stanford University, Stanford, California, 1983.

  20. 20.

    Gill, P. E., Murray, W., Saunders, M. A., andWright, M. H.,Maintaining LU Factors of a General Sparse Matrix, Linear Algebra and Its Applications, Vol. 88/89, pp. 239–270, 1987.

  21. 21.

    Powell, M. J. D.,An Error Growth in the Bartels-Golub and Fletcher-Matthews Algorithms for Updating Matrix Factorizations, Linear Algebra and Its Applications, Vol. 88/89, pp. 597–621, 1987.

  22. 22.

    Winkler, C.,Basis Factorization for Block Angular Linear Programs: Unified Theory of Partitioning and Decomposition Using the Simplex Method, Research Report RR-74-22, IIASA, Laxenburg, Austria, 1974.

  23. 23.

    Gille, P., andLoute, E.,Updating the LU Gaussian Decomposition for Rank-One Corrections: Application to Linear Programming Basis Partitioning Techniques, Cahier No. 8201, Faculte Universitaires Saint-Louis, Bruxelles, Belgium, 1982.

  24. 24.

    Fletcher, R., andHall, J. A. J.,Toward Reliable Linear Programming, Working Paper NA/120, Department of Mathematics and Computer Science, University of Dundee, Dundee, Scotland, 1989.

  25. 25.

    Lustig, I. J., Marsten, R., andShanno, D. I.,Computational Experience with a Primal-Dual Interior-Point Method for Linear Programming, Linear Algebra and Its Applications, Vol. 152, pp. 191–222, 1991.

  26. 26.

    Lustig, I. J., Mulvey, J. M., andCarpenter, T. J.,Formulating Two-Stage Stochastic Programs for Interior-Point Methods, Operations Research, Vol. 39, pp. 757–770, 1991.

  27. 27.

    Bisschop, J., andMeeraus, A.,Matrix Augmentation and the Partitioning in the Updating of the Basis Inverse, Mathematical Programming, Vol. 13, pp. 241–254, 1977.

  28. 28.

    Cottle, R. W.,Manifestations of the Schur Complement, Linear Algebra and Its Applications, Vol. 8, pp. 189–211, 1974.

  29. 29.

    Daniel, J. W., Gragg, W. B., Kaufman, L. C., andStewart, G. W.,Reorthogonalization and Stable Algorithms for Updating the Gram-Schmidt QR Factorization, Mathematics of Computation, Vol. 30, pp. 772–795, 1976.

  30. 30.

    Fletcher, R., andMatthews, F. P. J.,Stable Modification of Explicit LU Factors for Simplex Updates, Mathematical Programming, Vol. 30, pp. 267–284, 1984.

  31. 31.

    Gondzio, J.,Specialized Methods for Solving Multistage Linear Programming Problems, PhD Thesis, Department of Electronics, Warsaw University of Technology, Warsaw, Poland, 1988 (in Polish).

  32. 32.

    Gondzio, J.,Stable Algorithm for Updating Dense LU Factorization after Row or Column Exchange and Row and Column Addition or Deletion, Optimization, Vol. 23, pp. 7–26, 1992.

  33. 33.

    Erisman, A. M., Grimes, R. G., Lewis, J. D., Poole, W. G., Jr., andSimon, H. D.,Evaluation of Orderings for Unsymmetric Sparse Matrices, SIAM Journal on Scientific and Statistical Computations, Vol. 8, pp. 600–624, 1987.

  34. 34.

    Hellerman, E., andRarick, D. C.,Reinversion with the Preassigned Pivot Procedure, Mathematical Programming, Vol. 1, pp. 195–216, 1971.

  35. 35.

    McCarthy, J.,The Unit Hydrograph and Flood Routing, Conference of the North-Atlantic Division, US Corps of Engineers, Providence, Rhode Island, 1939.

  36. 36.

    Marsten, R.,The Design of the XMP Linear Programming Library, ACM Transactions on Mathematical Software, Vol. 7, pp. 481–497, 1981.

  37. 37.

    Gondzio, J.,Simplex Modifications Exploiting Special Features of Dynamic and Stochastic Dynamic Linear Programming Problems, Control and Cybernetics, Vol. 17, pp. 337–349, 1988.

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This research was supported by Programs CPBP02.15 and RPI.02.

Communicated by O. L. Mangasarian

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Gondzio, J., Ruszczyński, A. Sensitivity method for basis inverse representation in multistage stochastic linear programming problems. J Optim Theory Appl 74, 221–242 (1992). https://doi.org/10.1007/BF00940892

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Key Words

  • Linear programming
  • stochastic programming
  • simplex method