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Zonotopes and the LP-Newton method

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Abstract

Although linear programming problems can be solved in polynomial time by the ellipsoid method and interior-point algorithms, there still remains a long-standing open problem of devising a strongly polynomial algorithm for linear programming (or of disproving the existence of such an algorithm). The present work is motivated by an attempt toward solving this problem.

Linear programming problems can be formulated in terms of a zonotope, a kind of greedy polyhedron, on which linear optimization is made easily. We propose a method, called the LP-Newton method, for linear programming that is based on the zonotope formulation and the minimum-norm-point algorithm of Philip Wolfe. The LP-Newton method is a finite algorithm even for real-number input data with exact arithmetic computations. We show some preliminary computational results to examine the behavior of the LP-Newton method.

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Author notes

  1. Takumi Hayashi

    Present address: Nomura Securities Co. Ltd., Tokyo, Japan

Authors and Affiliations

  1. Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan

    Satoru Fujishige, Takumi Hayashi & Kei Yamashita

  2. Institut für Mathematische Optimierung, Technische Universität Braunschweig, 38106, Braunschweig, Germany

    Uwe Zimmermann

Authors
  1. Satoru Fujishige
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  2. Takumi Hayashi
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  3. Kei Yamashita
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  4. Uwe Zimmermann
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Corresponding author

Correspondence to Satoru Fujishige.

Additional information

Major part of this paper was presented as a plenary talk with the same title at ICOTA7 (December 12–15, 2007, Kobe, Japan) by the first author. The fourth author’s research was carried out while visiting RIMS in August 2007.

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Fujishige, S., Hayashi, T., Yamashita, K. et al. Zonotopes and the LP-Newton method. Optim Eng 10, 193–205 (2009). https://doi.org/10.1007/s11081-008-9067-x

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  • DOI: https://doi.org/10.1007/s11081-008-9067-x

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