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Theoretical convergence of large-step primal—dual interior point algorithms for linear programming

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Abstract

This paper proposes two sets of rules, Rule G and Rule P, for controlling step lengths in a generic primal—dual interior point method for solving the linear programming problem in standard form and its dual. Theoretically, Rule G ensures the global convergence, while Rule P, which is a special case of Rule G, ensures the O(nL) iteration polynomial-time computational complexity. Both rules depend only on the lengths of the steps from the current iterates in the primal and dual spaces to the respective boundaries of the primal and dual feasible regions. They rely neither on neighborhoods of the central trajectory nor on potential function. These rules allow large steps without performing any line search. Rule G is especially flexible enough for implementation in practically efficient primal—dual interior point algorithms.

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Author information

Authors and Affiliations

  1. Department of Information Sciences, Tokyo Institute of Technology, O-okayama, Meguro-ku, 152, Tokyo, Japan

    Masakazu Kojima

  2. IBM Almaden Research Center, San Jose, CA, USA

    Nimrod Megiddo

  3. School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel

    Nimrod Megiddo

  4. The Institute of Statistical Mathematics, Tokyo, Japan

    Shinji Mizuno

Authors
  1. Masakazu Kojima
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  2. Nimrod Megiddo
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  3. Shinji Mizuno
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Additional information

Part of the research was done when M. Kojima and S. Mizuno visited at the IBM Almaden Research Center. Partial support from the Office of Naval Research under Contracts N00014-87-C-0820 and N00014-91-C-0026 is acknowledged.

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Kojima, M., Megiddo, N. & Mizuno, S. Theoretical convergence of large-step primal—dual interior point algorithms for linear programming. Mathematical Programming 59, 1–21 (1993). https://doi.org/10.1007/BF01581234

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  • DOI: https://doi.org/10.1007/BF01581234

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