Advertisement

A Sensitivity Method for Solving Multistage Stochastic Linear Programming Problems

  • Jacek Gondzio
  • Andrzej Ruszczynski
Part of the Lecture Notes in Economics and Mathematical Systems book series (LNE, volume 331)

Abstract

A version of the simplex method for solving stochastic linear control problems is presented. The method takes advantage of the structure of the problem to achieve utmost memory economy in both data representation and basis inverse management.

Keywords

Simplex Method Sensitivity Matrix Multistage Stochastic Program Basis Inverse General Sparse Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bartels, R. H. and Golub, G. H. (1969). The simplex method of linear programming using LU decomposition. Communication on ACM 12, pp. 266–268.CrossRefGoogle Scholar
  2. Bisschop, J. and Meeraus, A. (1977). Matrix augmentation and the partitioning in the updating of the basis inverse. Mathematical Programming 13, pp. 241–254.CrossRefGoogle Scholar
  3. Bisschop, J. and Meeraus, A. (1980). Matrix augmentation and structure preservation in linearly constrained control problems. Mathematical Programming 18, pp. 7–15.CrossRefGoogle Scholar
  4. Daniel, J. W., Gragg, W. B., Kaufman, L. and Stewart, G. W. (1976). Reorthogonalization and stable algorithms for updating the Gram—Schmidt QR factorization. Mathematics of Computation 30, pp. 772–795.Google Scholar
  5. Fletcher, R. and Matthews, F. P. J. (1984). Stable modification of explicit LU factors for simplex updates. Mathematical Programming 30, pp. 267–284.CrossRefGoogle Scholar
  6. Fourer, R. (1982). Solving staircase linear programs by the simplex method, 1: inversion. Mathematical Programming 23, pp. 274–313.CrossRefGoogle Scholar
  7. Fourer, R. (1983). Solving staircase linear programs by the simplex method, 2: pricing. Mathematical Programming 25, pp. 251–292.CrossRefGoogle Scholar
  8. Gill, P. E., Murray, W., Saunders, M. A. and Wright, M. H. (1987). Maintaining LU factors of a general sparse matrix. Linear Algebra and its Applications 88 /89, pp. 239–270.CrossRefGoogle Scholar
  9. Gondzio, J. (1988a). Stable variant of the simplex method for solving supersparse linear programs. 3rd International Symposium on Systems Analysis and Simulation, Berlin 1988.Google Scholar
  10. Gondzio, J. (1988b). Simplex modifications exploiting special features of dynamic and stochastic dynamic linear programming problems. Control and Cybernetics, 1988 (to appear).Google Scholar
  11. Gondzio, J. and Ruszczynski, A. (1986). A package for solving dynamic linear programs. Institute of Automatic Control, Warsaw University of Technology, 1986 (in Polish).Google Scholar
  12. Murtagh, B. (1981). Advanced Linear Programming. McGraw—Hill, 1981.Google Scholar
  13. Murtagh, B. and Saunders, M. (1983). MINOS 5.0. User’s guide. System Optimization Laboratory, Stanford University, 1983.Google Scholar
  14. Powell, M. J. D. (1987). An error growth in the Bartels—Golub and Fletcher—Matthews algorithms for updating matrix factorizations. Linear Algebra and its Applications 88 /89, pp. 597–621.CrossRefGoogle Scholar
  15. Perold, A. F. and Dantzig, G. B. (1979). A basis factorization method for block triangular linear programs. in: Duff, I. S. and Stewart G. W. eds., Sparse Matrix Proceedings 1978, SIAM, Philadelphia, pp. 283–312.Google Scholar
  16. Reid, J. (1982). A sparsity—exploiting variant of the Bartels—Golub decomposition for linear programming bases. Mathematical Programming 24, pp. 55–69.CrossRefGoogle Scholar
  17. Wets, R. J.—B. (1986). Large scale linear programming techniques in stochastic programming. in: Ermoliev, Y. and Wets R. J.—B. (eds), Numerical Methods in Stochastic Programming, Springer—Verlag, Berlin 1986.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • Jacek Gondzio
    • 1
  • Andrzej Ruszczynski
    • 2
  1. 1.Systems Research InstitutePolish Academy of SciencesWarsawPoland
  2. 2.Institute of Automatic ControlWarsaw University of TechnologyPoland

Personalised recommendations