Abstract
A simple Newton-like descent algorithm for linear programming is proposed together with results of preliminary computational experiments on small- and medium-size problems. The proposed algorithm gives local superlinear convergence to the optimum and, experimentally, shows global linear convergence. It is similar to Karmarkar's algorithm in that it is an interior feasible direction method and self-correcting, while it is quite different from Karmarkar's in that it gives superlinear convergence and that no artificial extra constraint is introduced nor is protective geometry needed, but only affine geometry suffices.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
D. Avis and V. Chvátal, Note on Bland's pivoting rule,Math. Programming Stud.,8 (1978), 24–34.
M. Iri, Another “simple and fast” algorithm for linear programming, paper presented at the 12th International Symposium on Mathematical Programming, August 5–9, 1985, MIT, Boston.
M. Iri and H. Imai, A method of solving linear programming — with reference to the Karmarkar method and the penalty function method, Research Meeting of the Mathematical Programming Research Group of the Operations Research Society of Japan, February 16, 1985.
N. Karmarkar, A new polynomial-time algorithm for linear programming,Combinatorica,4 (1984), 373–395.
L. G. Khachian, A polynomial algorithm in linear programming,Dokl. Akad. Nauk SSSR,244 (1979), 1093–1096 (in Russian); transl. inSoviet Math. Dokl.,20 (1979), 191–194.
L. G. Khachian, Polynomial algorithms in linear programming,Zh. Vychisl. Mat. i Mat. Fiz.,20 (1980), 51–68 (in Russian); transl. inU.S.S.R. Comput. Math. and Math. Phys.,20 (1980), 53–72.
V. Klee and G. J. Minty, How good is the simplex algorithm?, inInequalities III (O. Shisha, ed.), Academic Press, New York, 1972, pp. 159–175.
Author information
Authors and Affiliations
Additional information
Communicated by Nimrod Megiddo.
The works of the first author and the second were supported in part by the Grant-in-Aid for Scientific Research (B) 60460130 (1985) and by the Grant-in-Aid for Encouragement of Young Scientists (A) 60790046 (1985), respectively, of the Ministry of Education, Science and Culture of Japan.
Rights and permissions
About this article
Cite this article
Iri, M., Imai, H. A multiplicative barrier function method for linear programming. Algorithmica 1, 455–482 (1986). https://doi.org/10.1007/BF01840457
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01840457