Annals of Operations Research

, Volume 22, Issue 1, pp 71–100 | Cite as

Vector processing in simplex and interior methods for linear programming

  • J. J. H. Forrest
  • J. A. Tomlin


We discuss the application of vector processing to various phases of simplex and interior point methods for linear programming. Preliminary computational results of experiments on the IBM 3090 vector facility will be presented.


Computational Result Interior Point Point Method Interior Point Method Vector Processing 


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  1. [1]
    I. Adler, M.G.C. Resende and G. Veiga, An implementation of Karmarkar's algorithm for linear programming, Report ORC 86-8, Dept. of IE/OR, University of California, Berkeley, CA (1986).Google Scholar
  2. [2]
    I. Adler, N. Karmarkar, M.G.C. Resende and G. Veiga, Data structures and programming techniques for the implementation of Karmarkar's algorithm, Technical Report, Dept. of IE/OR, University of California, Berkeley, CA (1987).Google Scholar
  3. [3]
    C.C. Ashcraft, R.G. Grimes, J.G. Lewis, B.W. Peyton and H.D. Simon, Progress in sparse matrix methods for large linear systems on vector supercomputers, Int. J. Supercomputer Appl. 1(1987)10–30.CrossRefGoogle Scholar
  4. [4]
    E.M.L. Beale, Advanced algorithmic features for general mathematical programming systems, in:Integer and Nonlinear Programming, ed. J. Abadie (North-Holland, Amsterdam, 1970) pp. 119–137.Google Scholar
  5. [5]
    M. Benichou, J.M. Gauthier, G. Hentges and G. Ribière, The efficient solution of large-scale linear programming problems — some algorithmic techniques and computational results, Math. Progr. 13(1977)280–322.CrossRefGoogle Scholar
  6. [6]
    W. Buchholz, The IBM System/370 vector architecture, IBM Systems J. 25(1986)51–62.Google Scholar
  7. [7]
    D. Carstens, Parallel processing for large scale linear programming and other application programs, Paper presented at the ORSA/TIMS Joint National Meeting, Los Angeles, CA (1978).Google Scholar
  8. [8]
    R.S. Clark and T.L. Wilson, Vector system performance of the IBM 3090, IBM System J. 25(1986)63–82.CrossRefGoogle Scholar
  9. [9]
    G.B. Dantzig,Linear Programming and Extensions (Princeton University Press, Princeton, NJ, 1963).Google Scholar
  10. [10]
    J.J. Dongarra, Designing algorithms for dense linear algebra problems on high performance computers, Paper presented at the Workshop on Supercomputers and Large Scale Optimization, University of Minnesota (1988).Google Scholar
  11. [11]
    J.J. Dongarra, F.G. Gustavson and A. Karp, Implementing linear algebra algorithms for dense matrices on a vector pipeline machine, SIAM Rev. 26(1984)91–112.CrossRefGoogle Scholar
  12. [12]
    I.S. Duff, Parallel implementation of multifrontal schemes, Parallel Computing 3(1986)193–204.CrossRefMathSciNetGoogle Scholar
  13. [13]
    I.S. Duff, The use of vector and parallel computers in the solution of large sparse linear equations, AERE-R.12393, Harwell Laboratory, Oxfordshire, UK (1986).Google Scholar
  14. [14]
    I.S. Duff, A.M. Erisman and J.K. Reid,Direct Methods for Sparse Matrices (Oxford University Press, London, 1986).Google Scholar
  15. [15]
    I.S. Duff and J.K. Reid, The multifrontal solution of indefinite sparse symmetric linear systems, ACM Trans. Math. Softw. 9(1983)302–325.CrossRefGoogle Scholar
  16. [16]
    J.J.H. Forrest, Linear programming using IBM 3090 vector facility, Paper presented at the E.M.L. Beale Memorial Symp., The Royal Society, London (1987).Google Scholar
  17. [17]
    J.J.H. Forrest and J.A. Tomlin, Updating triangular factors of the basis to maintain sparsity in the product form simplex method, Math. Progr. 2(1972)263–278.CrossRefGoogle Scholar
  18. [18]
    K. Gallivan, W. Jalby and U. Meier, The use of BLAS3 in linear algebra on a parallel processor with a hierarchical memory, SIAM J. Sci. Stat. Comput. 8(1987)1079–1084.CrossRefGoogle Scholar
  19. [19]
    D.M. Gay, Massive memory buys little speed for complete, in-core sparse Cholesky factorization, Numerical Analysis Manuscript 88-04, AT&T Bell Laboratories, Murray Hill, NJ. Presented at the 25th ORSA/TIMS Joint Meeting, Washington, D.C. (1988).Google Scholar
  20. [20]
    J.A. George and J.W. Liu,Computer Solution of Large Sparse Positive Definite Systems (Prentice-Hall, Englewood Cliffs, NJ, 1981).Google Scholar
  21. [21]
    P.E. Gill, W. Murray, M.A. Saunders, J.A. Tomlin and M.H. Wright, On projected Newton barrier methods for linear programming and an equivalence to Karmarkar's projective method, Math. Progr. 36(1986)183–209.Google Scholar
  22. [22]
    D. Goldfarb and S. Mehrotra, Relaxed variants of Karmarkar's algorithm for linear programs with unknown optimal objective function value, Math. Progr. 40(1988)183–195.CrossRefGoogle Scholar
  23. [23]
    G.H. Golub and C.F. Van Loan,Matrix Computations (North Oxford Academic, Oxford, and John Hopkins Press, Baltimore, 1983).Google Scholar
  24. [24]
    F.G. Gustavson, W.M. Liniger and R.A. Willoughby, Symbolic generation of an optimal Crout algorithm for sparse systems of linear equations, J. ACM 17(1970)87–109.CrossRefGoogle Scholar
  25. [25]
    P.M.J. Harris, Pivot selection methods of the Devex LP code, Math. Progr. 5(1973)1–28.CrossRefGoogle Scholar
  26. [26]
    IBM Corporation, IBM system/370 vector operations. Publication no. SA22-7125-2 (1986).Google Scholar
  27. [27]
    IBM Corporation, Engineering and Scientific Subroutine Library, guide and reference. Publication no. SC23-0184-2 (1987).Google Scholar
  28. [28]
    C.H. Johnson and E.P. Willard, TIMPS/ASC — An MPS implementation on a pipeline computer, Paper presented at the 8th Int. Symp. on Mathematical Programming, Stanford, CA (1973).Google Scholar
  29. [29]
    J.E. Kalan, Machine inspired enhancements of the simplex algorithm, Technical Report CS75001-R, Virginia Polytechnic Institute, Blacksburg, VA (1975).Google Scholar
  30. [30]
    N. Karmarkar, A new polynomial-time algorithm for linear programming, Combinatorica 4(1984)373–395.Google Scholar
  31. [31]
    J.G. Lewis and H.D. Simon, The impact of hardware gather/scatter on sparse Gaussian elimination, Boeing Computer Services, Mathematics and Modeling, Technical Report ETA-TR-33, Seattle, WA (1986).Google Scholar
  32. [32]
    K. McShane, C. Monma and D. Shanno, An implementation of a primal-dual interior point method for linear programming, Technical Report, Rutgers University, New Brunswick, NJ (1988).Google Scholar
  33. [33]
    Wm. Orchard-Hays,Advanced Linear Programming Computing Techniques (McGraw-Hill, New York, 1968).Google Scholar
  34. [34]
    C.C. Paige and M.A. Saunders, LSQR: An algorithm for sparse linear equations and sparse least-squares, ACM Trans. on Math. Softw. 8(1982)43–71.CrossRefGoogle Scholar
  35. [35]
    C.E. Pfefferkorn and J.A. Tomlin, Design of a linear programming system for the ILLIAC IV, Technical Report SOL 76-8, Dept. of Operations Research, Stanford University, and Technical Report 5487, NASA-Ames Institute for Advanced Computation, Sunnyvale, CA (1976).Google Scholar
  36. [36]
    J.K. Reid, FORTRAN subroutines for handling sparse linear programming bases, Report AERE-R.8269, Harwell Laboratory, Oxfordshire, UK (1976).Google Scholar
  37. [37]
    J.K. Reid, A sparsity exploiting variant of the Bartels-Golub decomposition for linear programming bases, Math. Progr. 24(1982)55–69.CrossRefGoogle Scholar
  38. [38]
    U.H. Suhl and L. Aittoniemi, Computing sparseLU-factorizations for large-scale linear programming bases, Arbeitspapier Nr. 58/87, Fachbereich Wirtschaftswissenschaft, Angewandte Informatik, Freie Universität Berlin (1987).Google Scholar
  39. [39]
    J.A. Tomlin, A note on comparing simplex and interior methods for linear programming, in:Progress in Mathematical Programming, ed. N. Megiddo (Springer-Verlag, New York, 1988) pp. 91–103.Google Scholar
  40. [40]
    J.A. Tomlin and J.S. Welch, Implementing an interior point method in a mathematical programming system, Paper presented at the 22nd TIMS/ORSA Joint Meeting, Miami, FL (1986).Google Scholar

Copyright information

© J.C. Baltzer AG, Scientific Publishing Company 1990

Authors and Affiliations

  • J. J. H. Forrest
    • 1
  • J. A. Tomlin
    • 2
  1. 1.IBM Thomas J. Watson Research CenterYorktown HeightsUSA
  2. 2.IBM Almaden Research CenterSan JoseUSA

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