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Dual barrier functions with superfast rates of convergence for the linear programming problem

Published online by Cambridge University Press:  17 February 2009

M. R. Osborne
M. R. Osborne
Affiliation:
Department of Statistics, Research School of Social Sciences, Australian National University, G. P. O. Box 4, Canberra, A. C. T. 2601, Australia
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Abstract

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It is shown that barrier functions applied to the dual linear program can be modified to give multiplier estimates that converge to the solution of the primal problem. Newton's method is considered for implementing this approach and numerical results presented. It has been shown that there is a connection between these methods and Karmarkar's algorithm, but for the class of problems considered further improvements are still required before those methods become competitive with active set methods.

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 29 , Issue 1 , July 1987 , pp. 39 - 58
Copyright
Copyright © Australian Mathematical Society 1987

References

[1]Adler, I., Megiddo, N. and Todd, M. J., “New results on the average behaviour of simplex algorithms”, Bull. Amer. Math. Soc. 11 (1984), 378–382.CrossRefGoogle Scholar
[2]Eriksson, J., Algorithms for Entropy and Mathematical Programming, Linköping Studies in Science and Technology Dissertations No. 63, Linköping University, 1981.Google Scholar
[3]Eriksson, J., “An iterative primal-dual algorithm for linear programming”, Report LiTH-MATR-1985–10, Linköping University, 1985.Google Scholar
[4]Fiacco, A. V. and McCormick, G. P., Nonlinear Programming (Wiley, New York, 1968).Google Scholar
[5]Fletcher, R., Practical Methods of Optimization (Wiley, Chichester, 1981).Google Scholar
[6]Gill, P. E., Murray, W., Saunders, M. A., Tomlin, J. A. and Wrigt, Margaret, “On projective Newton barrier methods for linear programming and an equivalence to Karmarkar's projective method”, Report SOL 85–11, Department of Operations Research, Stanford University, 1985.Google Scholar
[7]Haimovich, M., “The simplex algorithm is very good!—on the expected number of pivot steps and related properteis of random linear programs”, preprint, Columbia University, 1983.Google Scholar
[8]Jittorntrum, K. and Osborne, M. R., “Trajectory analysis and extrapolation in barrier function methods”, J. Aust. Math. Soc. Ser. B 20 (1979), 352–369.CrossRefGoogle Scholar
[9]Jittorntrum, K. and Osborne, M. R., “A modified barrier function mthod with improved rate of convergence for degenerate problems”, J. Aust. Math. Soc. Ser. B 21 (1980), 305–329.CrossRefGoogle Scholar
[10]Karmarker, N., “A new polynomial-time algorithm for linear programming”, Combinatorica, 4 (1984), 373–395.CrossRefGoogle Scholar
[11]Luenberger, D. G., Optimization by Vector Space Methods (Wiley, New York, 1969).Google Scholar
[12]Osborne, M. R., “Topics in optimization”, Report CS–72–279. Computer Science Department. Stanford University, 1972.Google Scholar
[13]Osborne, M. R., Finite Algorithms in Optimization and Data Analysis (Wiley, Chichester, 1985).Google Scholar
[14]Wright, Margaret, “Numerical methods for nonlinearly constrained optimization”, Report CS–76–566, Computer Science Department, Stanford University, 1976.Google Scholar