Abstract
The one common feature of all practical large-scale mathematical programming problems is sparseness. This applies whether the constraints are linear or nonlinear and whether the variables are continuous or discrete. The simplex method retains its central role in mathematical programming systems because in addition to being a device for changing the set of independent variables as appropriate it is now also an efficient method of exploiting sparseness. Supporting facilities that are now widely available are generalized upper bounds, branch and bound methods for integer variables and special ordered sets.
The paper reviews these facilities, and comments on the solution of large-scale nonlinear programming problems using general mathematical programming systems.
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References
J. Abadie and J. Carpentier, “Generalization of the Wolfe reduced gradient method to the case of nonlinear constraints”, in: R. Fletcher, ed., Optimization (Academic Press, London and New York, 1969) pp. 37–47.
R.H. Bartels and G.H. Golub, “The simplex method of linear programming using LU decomposition”, Communications of the Association for Computing Machinery 12 (1969) 266–268 and 275–278.
E.M.L. Beale, Mathematical programming in practice (Pitmans, London, 1968).
E.M.L. Beale, “Advanced algorithmic features for general mathematical programming systems”, in: J. Abadie, ed., Integer and nonlinear programming (North-Holland, Amsterdam, 1970) pp. 119–137.
E.M.L. Beale, “A conjugate gradient method of approximation programming”, in: R.W. Cottle and J. Krarup, eds., Optimization methods for resource allocation (English Universities Press, London, 1974) pp. 261–277.
E.M.L. Beale and J.A. Tomlin, “Special facilities in a general mathematical programming system for non-convex problems using ordered sets of variables”, in: J. Lawrence, ed., Proceedings of the fifth international conference on operational research (Tavistock Publications, London, 1970) pp. 447–454.
E.M.L. Beale and J.A. Tomlin, “An integer programming approach to a class of combinatorial problems”, Mathematical Programming 3 (1972) 339–344.
M. Benichou, J.M. Gauthier, P. Girodet, G. Hentges, G. Ribiere and D. Vincent, “Experiments in mixed-integer linear programming”, Mathematical Programming 1 (1971) 76–94.
R. Breu and C.A. Burdet, “Branch and bound experiments in zero-one programming”, Mathematical Programming Study 2 (1974) 1–50.
G.B. Dantzig, “Maximization of a linear function of variables subject to linear inequalities”, in: T.C. Koopmans, ed., Activity analysis of production and allocation (Wiley, New York, 1951) ch. XXI.
G.B. Dantzig and W. Orchard Hays, “The product form for the inverse in the simplex method”, Mathematics of Computation 8 (1954) 64–67.
G.B. Dantzig and R. Van Slyke, ‘Generalized upper bounding techniques”, Journal of Computer System Science 1 (1968) 213–226.
J.J.H. Forrest and J.A. Tomlin, “Updated triangular factors of the basis to maintain sparsity in the product form simplex method”, Mathematical Programming 2 (1972) 263–278.
J.J.H. Forrest, J.P.H. Hirst and J.A. Tomlin, “Practical solution of large mixed integer programming problems with Umpire”, Management Science 20 (1974) 736–773.
A. Geoffrion and R.E. Marsten, “Integer programming algorithms: a framework and state-of-the-art survey”, Management Science 18 (1972) 465–491.
R.E. Griffith and R.A. Stewart, “A nonlinear programming technique for the optimization of continuous processing systems”, Management Science 7 (1961) 379–392.
P.M.J. Harris, “Pivot selection methods of the Devex LP code”, Mathematical Programming Study 4 (1975) 30–57 (this volume).
J.E. Kalan, “Aspects of large-scale in-core linear programming”, in: Proceedings of the 1971 annual conference of the ACM, Chicago, Ill., August 3–5, 1971, pp. 304–313.
H.M. Markowitz, “The elimination form of inverse and its application to linear programming”, Management Science 3 (1957) 255–269.
G. Mitra, “Investigation of some branch and bound strategies for the solution of mixed integer linear programs”, Mathematical Programming 4 (1973) 155–170.
W. Orchard Hays, Advanced linear programming computing techniques (McGraw Hill, New York, 1968).
W.W. White, “A status report on computing algorithms for mathematical programming”, Computing Surveys 5 (1973) 135–166.
H.P. Williams, ‘Experiments in the formulation of integer programming problems”, Mathematical Programming Study 2 (1974) 180–197.
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Beale, E.M.L. (1975). The current algorithmic scope of mathematical programming systems. In: Balinski, M.L., Hellerman, E. (eds) Computational Practice in Mathematical Programming. Mathematical Programming Studies, vol 4. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0120707
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DOI: https://doi.org/10.1007/BFb0120707
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