Abstract
We develop an information geometric approach to conic programming. Information geometry is a differential geometric framework specifically tailored to deal with convexity, naturally arising in information science including statistics, machine learning and signal processing etc. First we introduce an information geometric framework of conic programming. Then we focus on semidefinite and symmetric cone programs. Recently, we demonstrated that the number of iterations of Mizuno–Todd–Ye predictor–corrector primal–dual interior-point methods is (asymptotically) expressed with an integral over the central trajectory called “the curvature integral”. The number of iterations of the algorithm is approximated surprisingly well with the integral even for fairly large linear/semidefinite programs with thousands of variables. Here we prove that “the curvature integral” admits a rigorous differential geometric expression based on information geometry. We also obtain an interesting information geometric global theorem on the central trajectory for linear programs. Together with the numerical evidence in the aforementioned work, we claim that “the number of iterations of the interior-point algorithm is expressed as a differential geometric quantity.”
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Acknowledgements
Takashi Tsuchiya is supported in part with Grant-in-Aid for Scientific Research (B), 2007, 19360192 from the Japan Society for the Promotion of Sciences. The authors would like to thank the two anonymous referees and the associate editor for their constructive comments which have helped to improve the presentation of the paper. Takashi Tsuchiya dedicates this paper to his late wife Hiroko Tsuchiya who passed away in January 2010, with his deepest respect and gratitude for her love and devotion to the family.
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Communicated by Nobuo Yamashita.
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Kakihara, S., Ohara, A. & Tsuchiya, T. Information Geometry and Interior-Point Algorithms in Semidefinite Programs and Symmetric Cone Programs. J Optim Theory Appl 157, 749–780 (2013). https://doi.org/10.1007/s10957-012-0180-9
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DOI: https://doi.org/10.1007/s10957-012-0180-9