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A sparsity-exploiting variant of the Bartels-Golub decomposition for linear programming bases. (English) Zbl 0492.90050

MSC:
90C06 Large-scale problems in mathematical programming
65K05 Numerical mathematical programming methods
90C05 Linear programming
Software:
XMP
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References:
[1] R.H. Bartels, ”A stabilization of the simplex method”,Numerische Mathematik 16 (1971) 414–434. · Zbl 0197.43305
[2] E.M.L. Beale, ”Sparseness in linear programming”, in: J.K. Reid, ed.,Large sparse sets of linear equations (Academic Press, London, 1971) pp. 1–15.
[3] A.R. Curtis and J.K. Reid, ”On the automatic scaling of matrices for Gaussian elimination”,Journal of the Institute of Mathematics and its Applications 10 (1972) 118–124. · Zbl 0249.65026
[4] I.S. Duff and J.K. Reid, ”A comparison of sparsity orderings for obtaining a pivotal sequence in Gaussian elimination”,Journal of the Institute of Mathematics and its Applications 14 (1974) 281–291. · Zbl 0308.65021
[5] B. Ford, ed., ”Parameterization of the environment for transportable numerical software”,ACM Transactions on Mathematical Software 4 (1978) 100–103.
[6] J.J.H. Forrest and J.A. Tomlin, ”Updating triangular factors of the basis to maintain sparsity in the product form simplex method”,Mathematical Programming 2 (1972) 263–278. · Zbl 0288.90048
[7] D.M. Gay, ”On combining the schemes of Reid and Saunders for sparse LP bases”, in: I.S. Duff and G.W. Stewart, eds.,Sparse matrix proceedings 1978 (SIAM, Philadelphia, PA, 1979) pp. 313–334.
[8] D. Goldfarb, ”On the Bartels–Golub decomposition for linear programming bases”,Mathematical Programming 13 (1977) 272–279. · Zbl 0379.90070
[9] F.G. Gustavson, ”Some basic techniques for solving sparse systems of linear equations, in: D.J. Rose and R.A. Willoughby, eds.,Sparse matrices and their applications (Plenum Press, New York, 1972) pp. 41–52.
[10] E. Hellerman and D.C. Rarick, ”Reinversion with the preassigned pivot procedure”,Mathematical Programming 1 (1971) 195–216. · Zbl 0246.65022
[11] R.D. McBride, ”A bump triangular dynamic factorization algorithm for the simplex method”,Mathematical Programming 18 (1980) 49–61. · Zbl 0428.90039
[12] H.M. Markowitz, ”The elimination form of the inverse and its applications to linear programming”,Management Science 3 (1957) 255–269. · Zbl 0995.90592
[13] R.E. Marsten, ”The design of the XMP linear programming library”,ACM Transactions on Mathematical Software (1981).
[14] W. Orchard-Hays,Advanced linear-programming computing techniques (McGraw-Hill, New York, 1968). · Zbl 0995.90595
[15] A.F. Perold, ”A degeneracy exploiting LU factorization for the simplex method”,Mathematical Programming 19 (1980) 239–254. · Zbl 0442.90054
[16] J.K. Reid, ”A note on the stability of Gaussian elimination”,Journal of the Institute of Mathematics and its Applications 8 (1971) 374–375. · Zbl 0229.65030
[17] J.K. Reid, ”Sparse linear programming using the Bartels–Golub decomposition”, verbal presentation at VIII International Symposium on Mathematical Programming, Stanford, CA (1973).
[18] J.K. Reid, ”Fortran subroutines for handling sparse linear programming bases”, Report AERER.8269, Harwell (1976).
[19] M.A. Saunders, ”Large-scale linear programming using the Cholesky factorization”, Report STAN-CS-72-252, Department of Computer Science, Stanford University, Stanford, CA (1972).
[20] M.A. Saunders, ”The complexity of LU updating in the simplex method”, in: R.S. Anderssen and R.P. Brent, eds.,The complexity of computational problem solving (University Press, Queensland, 1976) pp. 214–230.
[21] J.A. Tomlin, ”Pivoting for size and sparsity in linear programming inversion routines”,Journal of the Institute of Mathematics and its Applications 10 (1972) 289–295. · Zbl 0249.90039
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