Abstract
This paper contrasts the recent polynomial algorithms for linear programming of Khachian and Karmarkar. We show that each requires the solution of a weighted least-squares subproblem at every iteration. By comparing these subproblems we obtain further insights into the two methods.
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References
I. Adler, N. K. Karmarkar, M. G. C. Resende and G. Veiga, “An Implementation of Karmarkar’s Algorithm for Linear Programming”, manuscript, Department of Industrial Engineering and Operations Research, University of California, Berkeley, California, 1986.
K. M. Anstreicher, “Analysis of Karmarkar’s Algorithm for Fractional Linear Programming”, manuscript, School of Organization and Management, Yale Univerity, New Haven, Connecticut, 1985.
R. G. Bland, D. Goldfarb and M. J. Todd, “The Ellipsoid Method: a Survey”, Operations Research, 29, 1039–1091, 1981.
B. P. Burrell and M. J. Todd, “The Ellipsoid Method Generates Dual Variables”, Mathematics of Operations Research, 10, 688–700, 1985.
D. Gay, “A Variant of Karmarkar’s Linear Programming Algorithm for Problems in Standard Form”, Numerical Manuscript 85-10, AT&T Bell Laboratories, Murray Hill, New Jersey, 1985.
P. E. Gill, W. Murray, M. A. Saunders, J. A. Tomlin and M. H. Wright, “On Projected Newton Barrier Methods for Linear Programming and an Equivalence to Karmark’s Projective Method”, Technical report SOL85-11, Department of Operations Research, Stanford University, California, to appear in Mathematical Programming.
C. Gonzaga, “A Conical Projection Algorithm for Linear Programming”, manuscript, Department of Electrical Engineering and Computer Science, Univerity of California, Berkeley, California, 1985.
M. Grötschel, L. Lovász and A. Schrijver, “The Ellipsoid Method and its Consequences in Combinatorial Optimization”, Combinatorica, 1, 169–197, 1981.
M. Grötschel, L. Lovàsz and A. Schrijver, The Ellipsoid Method and Combinatorial Optimization, Springer-Verlag, Berlin, 1986, to appear.
N. K. Karmarkar, “A New Polynomial-Time Algorithm for Linear Programming”, Proceedings of the 16th Annual ACM Symposium on the Theory of Computing, 302–311, 1984a.
N. K. Karmarkar, “A New Polynomial-Time Algorithm for Linear Programming”, Combinatorica, 4, 373–395, 1984b.
N. K. Karmarkar, “Further Developments in the New Polynomial-Time Algorithm for Linear Programming”, presented at the 12th International Symposium on Mathematical Programming, Boston, Massachusetts, 1985.
N. K. Karmarkar and L. P. Sinha, “Application of Karmarkar’s Algorithm to Overseas Telecommunications Facilities Planning”, presented at the 12th International Symposium on Mathematical Programming, Boston, Massachusetts, 1985.
L. G. Khachian, “A Polynomial Algorithm in Linear Programming”, Soviet Mathematics Doklady, 20, 191–194, 1979.
L. G. Khachian, “Polynomial Algorithms in Linear Programming”, USSR Computational Mathematics and Mathematical Physics, 20, 53–72, 1980.
A. Steger, “An Extension of Karmarkar’s Algorithm for Bounded Linear Programming Problems”, M. S. Thesis, SUNY at Stonybrook, New York, 1985.
M. J. Todd and B. P. Burrell, “An Extension of Karmarkar’s Algorithm for Linear Programming Using Dual Variables”, Technical Report 648, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, New York, 1985.
D. B. Yudin and A. S. Nemirovsky, “Informational complexity and Effective Methods of Solution for Convex Extremal Problem”, Matekon, 13, 24–45, 1977.
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© 1988 Springer-Verlag Berlin Heidelberg
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Todd, M.J. (1988). Polynomial Algorithms for Linear Programming. In: Eiselt, H.A., Pederzoli, G. (eds) Advances in Optimization and Control. Lecture Notes in Economics and Mathematical Systems, vol 302. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-46629-8_4
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DOI: https://doi.org/10.1007/978-3-642-46629-8_4
Publisher Name: Springer, Berlin, Heidelberg
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