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Annals of Operations Research

, Volume 58, Issue 2, pp 67–98 | Cite as

A logarithmic barrier cutting plane method for convex programming

  • D. den Hertog
  • J. Kaliski
  • C. Roos
  • T. Terlaky
Nonlinear Programming

Abstract

The paper presents a logarithmic barrier cutting plane algorithm for convex (possibly non-smooth, semi-infinite) programming. Most cutting plane methods, like that of Kelley, and Cheney and Goldstein, solve a linear approximation (localization) of the problem and then generate an additional cut to remove the linear program's optimal point. Other methods, like the “central cutting” plane methods of Elzinga-Moore and Goffin-Vial, calculate a center of the linear approximation and then adjust the level of the objective, or separate the current center from the feasible set. In contrast to these existing techniques, we develop a method which does not solve the linear relaxations to optimality, but rather stays in the interior of the feasible set. The iterates follow the central path of a linear relaxation, until the current iterate either leaves the feasible set or is too close to the boundary. When this occurs, a new cut is generated and the algorithm iterates. We use the tools developed by den Hertog, Roos and Terlaky to analyze the effect of adding and deleting constraints in long-step logarithmic barrier methods for linear programming. Finally, implementation issues and computational results are presented. The test problems come from the class of numerically difficult convex geometric and semi-infinite programming problems.

Keywords

Column generation convex programming cutting plane methods decomposition interior point method linear programming logarithmic barrier function nonsmooth optimization semi-infinite programming 

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Copyright information

© J.C. Baltzer AG, Science Publishers 1995

Authors and Affiliations

  • D. den Hertog
    • 1
  • J. Kaliski
    • 1
  • C. Roos
    • 1
  • T. Terlaky
    • 1
  1. 1.Faculty of Technical Mathematics and InformaticsDelft University of TechnologyDelftThe Netherlands

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