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A logarithmic barrier cutting plane method for convex programming

  • Nonlinear Programming
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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.

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Authors and Affiliations

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

    D. den Hertog, J. Kaliski, C. Roos & T. Terlaky

Authors
  1. D. den Hertog
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  2. J. Kaliski
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  3. C. Roos
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  4. T. Terlaky
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Additional information

This work was completed under the support of a research grant of SHELL.

On leave from the Eötvös University, Budapest, and partially supported by OTKA No. 2116.

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den Hertog, D., Kaliski, J., Roos, C. et al. A logarithmic barrier cutting plane method for convex programming. Ann Oper Res 58, 67–98 (1995). https://doi.org/10.1007/BF02032162

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