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Revised Dantzig-Wolfe decomposition for staircase-structured linear programs.(English)Zbl 0638.90067

Staircase structured linear programs arise naturally in the study of engineering economic systems. One general approach to solving such LP’s is the technique of nested decomposition of the primal or dual problem. The research described in this paper proposes a revised decomposition algorithm that incorporates knowledge of the structure of the staircase basis in forming the decomposed linear programs. Column proposals from the revised subproblems are shown to achieve maximum penetration against the master problem basis. The proposed algorithm resorts to the regular Dantzig-Wolfe subproblem to test for optimality. The algorithm is shown to be finite and is compared to the Abrahamson-Wittrock algorithm. Computational results indicate substantial improvement over the Dantzig- Wolfe algorithm in most cases. A numerical example of the algorithm is provided in the appendix.

MSC:

90C06 Large-scale problems in mathematical programming
49M27 Decomposition methods
65K05 Numerical mathematical programming methods
90C05 Linear programming

Software:

MINOS

Cite

References:

[1]P.G. Abrahamson, ”A nested decomposition approach for solving staircase-structured linear programs,” in: G.B. Dantzig, M.A.H. Dempster and M. Kallio, eds.,Large-scale Linear Programming: Proceedings of a HASA workshop (International Institute for Applied Systems Analysis, Laxenburg, 1981) pp. 367–381. ·Zbl 0538.90052
[2]J.F. Benders, ”Partitioning procedures for solving mixed-variables programming problems,”Numerische Mathematik 4 (1962) 238–252. ·Zbl 0109.38302 ·doi:10.1007/BF01386316
[3]R.H. Cobb and J. Cord, ”Decomposition approaches for solving linked problems,” in: H.W. Kuhn, ed.,Proceedings of the Princeton Symposium on Mathematical Programming (Princeton University Press, Princeton, NJ, 1970) pp. 37–49. ·Zbl 0234.90031
[4]G.B. Dantzing,Linear Programming and Extensions (Princeton University Press, Princeton, NJ, 1963).
[5]G.B. Dantzig, ”Time-staged linear programs,” Technical Report SOL 80-28, Systems Optimization Laboratory, Department of Operations Research, Stanford University (Stanford, CA, 1980).
[6]G.B. Dantzig and P. Wolfe, ”Decomposition principle for linear programs,”Operations Research 8 (1960) 101–110. ·Zbl 0093.32806 ·doi:10.1287/opre.8.1.101
[7]G.B. Dantzig and P. Wolfe, ”The decomposition algorithm for linear programs,”Econometrica 29 (1961) 767–778. ·Zbl 0104.14305 ·doi:10.2307/1911818
[8]R. Fourer, ”Sparse Gaussian elimination of staircase linear systems,” Technical Report SOL 79-17, Systems Optimization Laboratory, Department of Operations Research, Stanford University (Stanford, CA, 1979).
[9]R. Fourer, ”Solving staircase linear programs by the simplex method, 1: Inversion,”Mathematical Programming 23 (1982) 274–313. ·Zbl 0487.90076 ·doi:10.1007/BF01583795
[10]R. Fourer, ”Solving staircase linear programs by the simplex method, 2: Pricing,”Mathematical Programming 25 (1983) 251–292. ·Zbl 0506.90054 ·doi:10.1007/BF02594780
[11]A.M. Geoffrion, ”Elements of large-scale mathematical programming, part I: Concepts”,Management Science 16 (1970) 653–675. ·Zbl 0209.22801
[12]C.R. Glassey, ”Dynamic linear programs for production scheduling,”Operations Research 19 (1971) 45–56. ·Zbl 0216.26702 ·doi:10.1287/opre.19.1.45
[13]C.R. Glassey, ”Nested decomposition and multi-stage linear programs,”Management Science 20 (1973) 282–292. ·Zbl 0313.90037 ·doi:10.1287/mnsc.20.3.282
[14]J.K. Ho and E. Loute, ”A comparative study of two methods for staircase linear programs,”Transactions on Mathematical Software, ACM 6 (1980) 17–30. ·Zbl 0432.90048 ·doi:10.1145/355873.355875
[15]J.K. Ho and E. Loute, ”A set of staircase linear programming test problems,”Mathematical Programming 20 (1981) 245–250. ·Zbl 0448.90036 ·doi:10.1007/BF01589349
[16]J.K. Ho and E. Loute, ”An advanced implementation of the Dantzig-Wolfe decomposition algorithm for linear programming,”Mathematical Programming 20 (1981) 303–326. ·Zbl 0468.90042 ·doi:10.1007/BF01589355
[17]J.K. Ho and E. Loute, ”Computational experience with advanced implementation of decomposition algorithms for linear programming,”Mathematical Programming 27 (1983) 283–290. ·Zbl 0521.65043 ·doi:10.1007/BF02591904
[18]J.K. Ho and A.S. Manne, ”Nested decomposition for dynamic models,”Mathematical Programming 6 (1974) 121–140. ·Zbl 0294.90051 ·doi:10.1007/BF01580231
[19]M. Kallio and E.L. Porteus, ”Decomposition of arborescent linear programs,”Mathematical Programming 13 (1973) 348–356. ·Zbl 0377.90068 ·doi:10.1007/BF01584347
[20]L.S. Lasdon,Optimization Theory for Large Systems (Macmillan, New York, 1970). ·Zbl 0224.90038
[21]D.F. Lynch, ”A nested decomposition algorithm with surrogate rows for staircase structured linear programs,” M.S. thesis, School of Operations Research and Industrial Engineering, Cornell University (Ithaca, NY 1983).
[22]D.F. Lynch, ”The guided decomposition algorithm for linear programs,” Ph.D. thesis, School of Operations Research and Industrial Engineering, Cornell University (Ithaca, NY, 1984).
[23]A.S. Manne, ”Sufficient conditions for optimality in an infinite horizon development plan,”Econometrica 38 (1970) 18–38. ·doi:10.2307/1909238
[24]B.A. Murtagh and M.A. Saunders, ”MINOS user’s guide,” Technical Report SOL 77-9, Systems Optimization Laboratory, Department of Operations Research, Stanford University (Stanford, CA, 1977).
[25]G.L. Nemhauser, ”Decomposition of linear programs by dynamic programming,”Naval Research Logistics Quarterly 11 (1974) 191–196. ·Zbl 0136.14105 ·doi:10.1002/nav.3800110206
[26]P.V. Preckel, ”Modules for use with MINOS/AUGMENTED in solving sequences of mathematical programs,” Technical Report SOL 80-15, Systems Optimization Laboratory, Department of Operations Research, Stanford University (Stanford, CA, 1980).
[27]M.A. Saunders, ”MINOS system manual,” Technical Report SOL 77-31, Systems Optimization Laboratory, Department of Operations Research, Stanford University (Stanford, CA, 1977).
[28]R.M. van Slyk and R. Wets, ”L-shaped linear programs with applications to optimal control and stochastic programming,”SIAM Journal of Applied Mathematics 17 (1969) 638–663. ·Zbl 0197.45602 ·doi:10.1137/0117061
[29]R.J. Wittrock, ”Advances in a nested decomposition algorithm for solving staircase linear programs,” Technical Report SOL 83-2, Systems Optimization Laboratory, Department of Operations Research, Stanford University (Stanford, CA, 1983).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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