Abstract
We present a modification of Karmarkar's linear programming algorithm. Our algorithm uses a recentered projected gradient approach thereby obviatinga priori knowledge of the optimal objective function value. Assuming primal and dual nondegeneracy, we prove that our algorithm converges. We present computational comparisons between our algorithm and the revised simplex method. For small, dense constraint matrices we saw little difference between the two methods.
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L. V. Atkinson and P. J. Harley,An Introduction to Numerical Methods with Pascal, Addison-Wesley, Reading, MA, 1983.
E. R. Barnes, A variation on Karmarkar's algorithm for solving linear programming problems, Manuscript, IBM T. J. Watson Research Center, Yorktown Heights, NY, 1985.
T. M. Cavalier and A. L. Soyster, Some computational experience and a modification of the Karmarkar algorithm, ISME Working Paper 85–105, The Pennsylvania State University, 1985.
V. Chvátal,Linear Programming, Freeman, New York and San Francisco, 1983.
S. C. Eisenstat, Efficient implementation of a class of preconditioned conjugate gradient methods,SIAM J. Sci. Statist. Comput.,2 (1985), 4–7.
N. Karmarkar, A new polynomial-time algorithm for linear programming,Combinatorica,4 (1984), 373–395.
L. S. Lasdon,Optimization Theory for Large Systems, Macmillan, New York, 1970.
N. Megiddo, A variation on Karmarkar's algorithm, Manuscript, IBM Research Laboratory, San Jose, CA, 1985.
M. J. Todd and B. P. Burrell, An extension of Karmarkar's algorithm for linear programming using dual variables, Technical Report No. 648, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY, 1985.
J. A. Tomlin, An experimental approach to Karmarkar's projective method for linear programming, Manuscript, Ketron Inc., Mountain View, CA, 1985.
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Communicated by Nimrod Megiddo.
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Vanderbei, R.J., Meketon, M.S. & Freedman, B.A. A modification of karmarkar's linear programming algorithm. Algorithmica 1, 395–407 (1986). https://doi.org/10.1007/BF01840454
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DOI: https://doi.org/10.1007/BF01840454