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Curvature integrals and iteration complexities in SDP and symmetric cone programs

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  • Satoshi Kakihara
  • Atsumi Ohara
  • Takashi Tsuchiya

Abstract

In this paper, we study iteration complexities of Mizuno-Todd-Ye predictor-corrector (MTY-PC) algorithms in SDP and symmetric cone programs by way of curvature integrals. The curvature integral is defined along the central path, reflecting the geometric structure of the central path. Integrating curvature along the central path, we obtain a precise estimate of the number of iterations to solve the problem. It has been shown for LP that the number of iterations is asymptotically precisely estimated with the integral divided by $\sqrt{\beta}$ , where β is the opening parameter of the neighborhood of the central path in MTY-PC algorithms. Furthermore, this estimate agrees quite well with the observed number of iterations of the algorithm even when β is close to one and when applied to solve large LP instances from NETLIB. The purpose of this paper is to develop direct extensions of these two results to SDP and symmetric cone programs. More specifically, we give concrete formulas for curvature integrals in SDP and symmetric cone programs and give asymptotic estimates for iteration complexities. Through numerical experiments with large SDP instances from SDPLIB, we demonstrate that the number of iterations is explained quite well with the integral even for a large step size which is enough to solve practical large problems. Copyright Springer Science+Business Media New York 2014

Suggested Citation

  • Satoshi Kakihara & Atsumi Ohara & Takashi Tsuchiya, 2014. "Curvature integrals and iteration complexities in SDP and symmetric cone programs," Computational Optimization and Applications, Springer, vol. 57(3), pages 623-665, April.
    • Handle: RePEc:spr:coopap:v:57:y:2014:i:3:p:623-665
    DOI: 10.1007/s10589-013-9608-x
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    References listed on IDEAS

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    1. Satoshi Kakihara & Atsumi Ohara & Takashi Tsuchiya, 2013. "Information Geometry and Interior-Point Algorithms in Semidefinite Programs and Symmetric Cone Programs," Journal of Optimization Theory and Applications, Springer, vol. 157(3), pages 749-780, June.
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    Cited by:

    1. M. Sayadi Shahraki & H. Mansouri & M. Zangiabadi, 2017. "Two wide neighborhood interior-point methods for symmetric cone optimization," Computational Optimization and Applications, Springer, vol. 68(1), pages 29-55, September.

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