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A polynomial newton method for linear programming

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Abstract

An algorithm is presented for solving a set of linear equations on the nonnegative orthant. This problem can be made equivalent to the maximization of a simple concave function subject to a similar set of linear equations and bounds on the variables. A Newton method can then be used which enforces a uniform lower bound which increases geometrically with the number of iterations. The basic steps are a projection operation and a simple line search. It is shown that this procedure either proves in at mostO(n 2 m 2 L) operations that there is no solution or, else, computes an exact solution in at mostO(n 3 m 2 L) operations.

The linear programming problem is treated as a parametrized feasibility problem and solved in at mostO(n 3 m 2 L) operations.

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Authors and Affiliations

  1. CORE, Université Catholique de Louvain, 34 voie du Roman Pays, B1348, Louvain-la-Neuve, Belgium

    Guy de Ghellinck & Jean -Philippe Vial

  2. Departement COMIN, Université de Genève, 2 rue de Candolle, C.H. 1211, Genève 4, Switzerland

    Jean -Philippe Vial

Authors
  1. Guy de Ghellinck
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  2. Jean -Philippe Vial
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Communicated by Nimrod Megiddo.

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de Ghellinck, G., Vial, J.P. A polynomial newton method for linear programming. Algorithmica 1, 425–453 (1986). https://doi.org/10.1007/BF01840456

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  • DOI: https://doi.org/10.1007/BF01840456

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