Abstract
Special methods for dealing with constraints of the formx j ≤ x k , called variable upper bounds, were introduced by Schrage. Here we describe a method that circumvents the massive degeneracy inherent in these constraints and show how it can be implemented using triangular basis factorizations.
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References
R.H. Bartels, “A stabilization of the simplex method”,Numerische Mathematik 16 (1971) 414–434.
R.H. Bartels and G.H. Golub, “The simplex method for linear programming usingLU decomposition”,Communications of the ACM 12 (1969) 266–268.
R.H. Bartels, G.H. Golub and M.A. Saunders, “Numerical techniques in mathematical programming”, in: J. Rosen, O. Mangasarian and K. Ritter, eds.,Nonlinear programming (Academic Press, New York, 1970) pp. 123–176.
R.C. Daniel, “A note on Schrage's generalized variable upper bounds”,Mathematical Programming 15 (1978) 349–351.
R. Dutton, G. Hinman and C.B. Millham, “The optimal location of nuclear-power facilities in the Pacific Northwest”,Operations Research 22 (1974) 478–487.
J.J.H. Forest and J.A. Tomlin, “Updating triangular factors of the basis to maintain sparsity in the product form of the simplex method”,Mathematical Programming 2 (1972) 263–278.
P. Gill and W. Murray, “A numerically stable form of the simplex algorithm”,Journal of Linear Algebra and its Applications 7 (1973) 99–138.
P. Gill, W. Murray and M.A. Saunders, “Methods for computing and modifying the LDV factors of a matrix”,Mathematics of Computation 29 (1975) 1051–1077.
F. Glover, “Compact LP bases for a class of IP problems”,Mathematical Programming 12 (1977) 102–109.
D. Goldfarb, “On the Bartels-Golub decomposition for linear programming bases”,Mathematical Programming 13 (1977) 272–279.
G. Roodman and L. Schwartz, “The dynamic plant location problem”, working paper, University of Rochester (1974).
M.A. Saunders, “Large-scale programming using the Cholesky factorization”, Report Stan-CS-72252, Computer Science Department, Stanford University (Stanford, 1972).
L. Schrage, “Implicit representation of variable upper bounds in linear programming”,Mathematical Programming Study 4 (1975) 118–132.
L. Schrage, “Implicit representation of generalized variable upper bounds in linear programming”,Mathematical Programming 14 (1978) 11–20.
M.J. Todd, “Traversing large pieces of linearity in algorithms that solve equations by following piecewise-linear paths”,Mathematics of Operations Research 5 (1980) 242–257.
M.J. Todd, “Numerical stability and sparsity in piecewise-linear algorithms”, in: S.M. Robinson, ed.,Analysis and computation of fixed points (Academic Press, New York, 1980) pp. 1–24.
P. Wolfe, “The composite simplex algorithm”,SIAM Review 7 (1965) 42–54.
H.P. Williams, “Experiments in the formulation of integer programming problems”,Mathematical Programming Study 2 (1974) 180–197.
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This research was partially supported by National Science Foundation Grant ECS-7921279 and by a Guggenheim fellowship.
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Todd, M.J. An implementation of the simplex method for linear programming problems with variable upper bounds. Mathematical Programming 23, 34–49 (1982). https://doi.org/10.1007/BF01583778
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DOI: https://doi.org/10.1007/BF01583778