Abstract
This paper describes some computational experience with projective methods for linear programming. Rather than compare a particular implementation of a projective algorithm with versions of the simplex method, our purpose is to identify significant characteristics and limitations of a specific class of projective methods. A number of variations are possible within this class, and this study includes a comparison of single-phase and two-phase versions.
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Anstreicher, K.M., A combined ‘phase I — phase II’ projective algorithm for linear programming, manuscript, Yale School of Organization and Management (1986), to appear in Mathematical Programming.
Ghellinck, G. de, Vial, J.-P., A polynomial Newton method for linear programming, Algorithmica 1, (1986), 425–453.
Ghellinck, G. de, Vial, J.-P., An extension of Karmarkar's algorithm for solving a system of linear homogeneous equations on the simplex, Mathematical Programming 39 (1987) 79–92.
Chu, E., George, A., Liu, J. and Ng, E., SPARSPAK: Waterloo Sparse Matrix Package, User's Guide for SPARSPAK-A, Research Report CS-84-36, Department of Computer Science, University of Waterloo (November 1984).
Dongarra, J.J., Bunch, J.R., Moler, C.B., Stewart, G.W., LIMPACK User's Guide, SIAM (1979).
Dongarra, J.J., Grosse, E., Distribution of mathematical software via electronic mail, Communications of the ACM, Vol. 30, No. 5 403–407.
Duff, I.S., Reid, J.K., MA-27-A set Fortran subroutines for solving sparse symmetric sets of linear equations, Report AERE R-10533, Computer Science and Systems Division, AERE Harwell, U.K. (1982).
Eisenstat, S.C., Gursky, M.C., Schultz, M.H., Sherman, A.H., Yale sparse matrix package I: The symmetric codes, International Journal of Numerical methods in Engineering 18 (1982), 1145–1151.
George, A., Ng, E., SPARSPAK: Waterloo sparse matrix package, User's Guide for SPARSPAK-B, Research Report CS-84-37, Department of Computer Science, University of Waterloo (November 1984).
Gill, P.E., Murray, W., Saunders, M.A., Wright M.H., Two steplength algorithms for numerical optimization, Technical report SOL 79-25, Systems optimization Laboratory, department of Operations Research, Stanford University (1979).
Gill, P.E., Murray, W., Wright M.H., Practical optimization, Academic Press (1981).
Gill, P.E., Murray, W., Saunders, M.A., Wright M.H., Users guide for NPSOL (Version 4.0): A Fortran package for nonlinear programming, technical report SOL 86-2, Systems Optimization Laboratory, Department of Operations Research, Stanford University (January 1986; revised Version 4.2 in 1987).
Gill, P. E., Murray, W., Saunders, M.A., Tomlin, J.A., Wright M.H., On projected Newton barrier methods for linear programming and an equivalence to Karmarkar's projective method, Mathematical Programming 36 (1986) 183–209.
Gill, P.E., Murray, W., Saunders, M.A., Wright M.H., A note on nonlinear approaches to linear programming, Technical Report SOL 86-7, Systems Optimization Laboratory, Department of Operations Research, Stanford University (April 1986).
Karmarkar, N., A new polynomial-time algorithm for linear programming, Combinatorica 4 (1984) 373–395.
Subroutine F04QAF, NAG Fortran Library Manual, Mark 12, Volume 5, Numerical Algorithms Group, Downers Grove, IL U.S.A., and Oxford, U.K. (March 1987).
Paige, C.C., Saunders, M.A., LSQR: An algorithm for sparse linear equations and sparse least-squares, ACM Transactions on Mathematical Software 8 (1982), 43–71.
SMPAK User's Guide Version 1.0, Scientific Computing Associates, U.S.A., (1985).
Todd, M.J., Burrell, B.P., An extension of Karmarkar's algorithm for linear programming using dual variables, Algorithmica 1 (1986) 409–424.
Todd, M.J., On Anstreicher's combined phase I-phase II projective algorithm for linear programming, Technical Report No.776, School of Operations Research and Industrial Engineering, Cornell University (1988).
Vial, J.-P., A unified approach to projective algorithms for linear programming, CORE Discussion Paper No 8747, Center for Operations Research and Econometrics, Université Catholique de Louvain (September 1987); revised version appears in these proceedings.
Ye, Y., Kojima, M., Recovering optimal dual solutions in Karmarkar's polynomial algorithm for linear programming, Mathematical Programming 39 (1987) 305–317.
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© 1989 Springer Verlag
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Fraley, C., Vial, JP. (1989). Numerical study of projective methods for linear programming. In: Dolecki, S. (eds) Optimization. Lecture Notes in Mathematics, vol 1405. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0083584
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DOI: https://doi.org/10.1007/BFb0083584
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