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Fractional differential equation approach for convex optimization with convergence rate analysis

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Abstract

A fractional differential equation (FDE) based algorithm for convex optimization is presented in this paper, which generalizes ordinary differential equation (ODE) based algorithm by providing an additional tunable parameter \(\alpha \in (0,1]\). The convergence of the algorithm is analyzed. For the strongly convex case, the algorithm achieves at least the Mittag-Leffler convergence, while for the general case, the algorithm achieves at least an \(O(1/t^\alpha )\) convergence rate. Numerical simulations show that the FDE based algorithm may have faster or slower convergence speed than the ODE based one, depending on specific problems.

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Acknowledgements

This research was supported in part by the U.S. Army Research Office under Grants W911NF-19-1-0176, W911NF-15-1-0218 and in part by the National Natural Science Foundation of China under Grant 61873024, and in part by Science and Technology on Space Intelligent Control Laboratory for National Defense under Grant KGJZDSYS-2018-13, and in part by the Fundamental Research Funds for the China Central Universities of USTB under Grant FRF-TP-17-088A1.

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Authors and Affiliations

  1. Key Laboratory of Knowledge Automation for Industrial Processes of Ministry of Education, School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing, 100083, People’s Republic of China

    Shu Liang

  2. Institute of Artificial Intelligence, University of Science and Technology Beijing, Beijing, 100083, People’s Republic of China

    Shu Liang

  3. Department of Electrical and Computer Engineering, Wayne State University, Detroit, MI, 48202, USA

    Leyi Wang

  4. Department of Mathmatics, Wayne State University, Detroit, MI, 48202, USA

    George Yin

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  1. Shu Liang
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  2. Leyi Wang
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  3. George Yin
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Correspondence to Shu Liang.

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Liang, S., Wang, L. & Yin, G. Fractional differential equation approach for convex optimization with convergence rate analysis. Optim Lett 14, 145–155 (2020). https://doi.org/10.1007/s11590-019-01437-6

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