# An exterior-point method for linear programming problems

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## Abstract

This paper proves the convergence of an algorithm for solving linear programming problems in*O(mn*^{2}) arithmetic operations. The method is called an exterior-point procedure, because it obtains a sequence of approximations falling outside the set*U* of feasible solutions. Each iteration consists of a single step within some constraining hyperplane, followed by one or more projections which force the new approximation to fall within some envelope about*U*. The paper also discusses several numerical applications. In some types of problems, the method is considerably faster than a standard simplex method program when the size of the problem is sufficiently large.

## Key Words

Linear programming polynomial time algorithms## Preview

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## References

- 1.Andrus, J. F.,
*An Exterior-Point Method for the Convex Programming Problem*, Journal of Optimization Theory and Applications, Vol. 72, pp. 37–63, 1992.CrossRefGoogle Scholar - 2.Gonzaga, C. C.,
*An Algorithm for Solving Linear Programming Problems in O(n*^{3}L)*Operations*, Progress in Mathematical Programming: Interior and Related Methods, Edited by N. Megiddo, Springer Verlag, New York, New York, pp. 1–28, 1989.Google Scholar - 3.Karmarkar, N.,
*A New Polynomial Time Algorithm for Linear Programming*, Combinatorica, Vol. 4, pp. 373–395, 1984.Google Scholar - 4.Khachian, L. G.,
*Polynomial Algorithms in Linear Programming*, USSR Computational Mathematics and Mathematical Physics, Vol. 20, pp. 53–72, 1980.CrossRefGoogle Scholar - 5.Monteiro, R. D. C., andAdler, I.,
*Interior Path Following Primal-Dual Algorithms, Part 1: Linear Programming*, Mathematical Programming, Vol. 44, pp. 27–41, 1989.CrossRefGoogle Scholar - 6.Renegar, J.,
*A Polynomial-Time Algorithm, Based on Newton's Method, for Linear Programming*, Mathematical Programming, Vol. 40, pp. 59–93, 1988.CrossRefGoogle Scholar - 7.Vaidya, P. M.,
*An Algorithm for Linear Programming Which Requires O(((m+n)n*^{2}+(*m+n*)^{1.5}*n)L*)*Arithmetic Operations*, Mathematical Programming, Vol. 47, pp. 175–201, 1990.CrossRefGoogle Scholar - 8.Domich, P. D., Hoffman, K. L., Jackson, R. H. F., Saunders, P. S., andShier, D. R.,
*Comparison of Mathematical Programming Software: A Case Study Using Discrete L*_{1}-*Approximation Codes*, Computers and Operations Research, Vol. 14, pp. 435–447, 1987.CrossRefGoogle Scholar

## Copyright information

© Plenum Publishing Corporation 1996