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New results on subgradient methods for strongly convex optimization problems with a unified analysis

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  • Masaru Ito

    (Tokyo Institute of Technology)

Abstract

We develop subgradient- and gradient-based methods for minimizing strongly convex functions under a notion which generalizes the standard Euclidean strong convexity. We propose a unifying framework for subgradient methods which yields two kinds of methods, namely, the proximal gradient method (PGM) and the conditional gradient method (CGM), unifying several existing methods. The unifying framework provides tools to analyze the convergence of PGMs and CGMs for non-smooth, (weakly) smooth, and further for structured problems such as the inexact oracle models. The proposed subgradient methods yield optimal PGMs for several classes of problems and yield optimal and nearly optimal CGMs for smooth and weakly smooth problems, respectively.

Suggested Citation

  • Masaru Ito, 2016. "New results on subgradient methods for strongly convex optimization problems with a unified analysis," Computational Optimization and Applications, Springer, vol. 65(1), pages 127-172, September.
    • Handle: RePEc:spr:coopap:v:65:y:2016:i:1:d:10.1007_s10589-016-9841-1
    DOI: 10.1007/s10589-016-9841-1
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    References listed on IDEAS

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    1. DEVOLDER, Olivier & GLINEUR, François & NESTEROV, Yurii, 2013. "First-order methods with inexact oracle: the strongly convex case," LIDAM Discussion Papers CORE 2013016, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. NESTEROV, Yurii, 2013. "Gradient methods for minimizing composite functions," LIDAM Reprints CORE 2510, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. DEVOLDER, Olivier & GLINEUR, François & NESTEROV, Yurii, 2011. "First-order methods of smooth convex optimization with inexact oracle," LIDAM Discussion Papers CORE 2011002, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. NESTEROV, Yu., 2005. "Excessive gap technique in nonsmooth convex minimization," LIDAM Reprints CORE 1818, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    5. NESTEROV, Yu., 2005. "Smooth minimization of non-smooth functions," LIDAM Reprints CORE 1819, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    6. NESTEROV, Yurii, 2015. "Universal gradient methods for convex optimization problems," LIDAM Reprints CORE 2701, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    7. NESTEROV, Yurii, 2015. "Complexity bounds for primal-dual methods minimizing the model of objective function," LIDAM Discussion Papers CORE 2015003, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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