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Primal-dual algorithms for linear programming based on the logarithmic barrier method

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Abstract

In this paper, we deal with primal-dual interior point methods for solving the linear programming problem. We present a short-step and a long-step path-following primal-dual method and derive polynomial-time bounds for both methods. The iteration bounds are as usual in the existing literature, namely\(O(\sqrt n L)\) iterations for the short-step variant andO(nL) for the long-step variant. In the analysis of both variants, we use a new proximity measure, which is closely related to the Euclidean norm of the scaled search direction vectors. The analysis of the long-step method depends strongly on the fact that the usual search directions form a descent direction for the so-called primal-dual logarithmic barrier function.

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Author information

Author notes

  1. T. Terlaky (Assistant Professor)

    Present address: Eötvös University, Budapest, Hungary

Authors and Affiliations

  1. Faculty of Technical Mathematics and Computer Science, Delft University of Technology, Delft, The Netherlands

    B. Jansen (PhD Student), C. Roos (Assistant Professor) & T. Terlaky (Assistant Professor)

  2. Department of Commercial and Industrial Economics, University of Geneva, Geneva, Switzerland

    J. P. Vial Professor

Authors
  1. B. Jansen
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  2. C. Roos
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  3. T. Terlaky
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  4. J. P. Vial Professor
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Additional information

Communicated by L. C. W. Dixon

This work was supported by a research grant from Shell, by the Dutch Organization for Scientific Research (NWO) Grant 611-304-028, by the Hungarian National Research Foundation Grant OTKA-2116, and by the Swiss National Foundation for Scientific Research Grant 12-26434.89.

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Jansen, B., Roos, C., Terlaky, T. et al. Primal-dual algorithms for linear programming based on the logarithmic barrier method. J Optim Theory Appl 83, 1–26 (1994). https://doi.org/10.1007/BF02191759

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