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Extension of the LP-Newton method to conic programming problems via semi-infinite representation

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Abstract

The LP-Newton method solves linear programming (LP) problems by repeatedly projecting a current point onto a certain relevant polytope. In this paper, we extend the algorithmic framework of the LP-Newton method to conic programming (CP) problems via a linear semi-infinite programming (LSIP) reformulation. In this extension, we produce a sequence by projection onto polyhedral cones constructed from LP problems obtained by finitely relaxing the LSIP problem equivalent to the CP problem. We show global convergence of the proposed algorithm under mild assumptions. To investigate its efficiency, we apply our proposed algorithm and a primal-dual interior-point method to second-order cone programming problems and compare the obtained results.

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Acknowledgments

We would like to thank the editor Claude Brezinski, the anonymous reviewers, and Dr. Bruno Figueira Lourenço for useful comments.

Funding

This research is supported by JST CREST JPMJCR14D2 and JSPS Grants-in-Aid for Young Scientists 15K15943, 16K16357, and 19K15247.

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Authors and Affiliations

  1. Department of Statistical Inference and Mathematics, The Institute of Statistical Mathematics, 10-3, Midori-cho, Tachikawa, Tokyo, 190-8562, Japan

    Mirai Tanaka

  2. RIKEN Center for Advanced Intelligence Project, 1-4-1 Nihonbashi, Chuo-ku, Tokyo, 103-0027, Japan

    Takayuki Okuno

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  1. Mirai Tanaka
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  2. Takayuki Okuno
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Correspondence to Mirai Tanaka.

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Tanaka, M., Okuno, T. Extension of the LP-Newton method to conic programming problems via semi-infinite representation. Numer Algor 86, 1285–1302 (2021). https://doi.org/10.1007/s11075-020-00933-6

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