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Global complexity analysis of inexact successive quadratic approximation methods for regularized optimization under mild assumptions

Author

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  • Wei Peng

    (Chinese Academy of Military Science)

  • Hui Zhang

    (National University of Defense Technology)

  • Xiaoya Zhang

    (Chinese Academy of Military Science)

  • Lizhi Cheng

    (National University of Defense Technology)

Abstract

Successive quadratic approximations (SQA) are numerically efficient for minimizing the sum of a smooth function and a convex function. The iteration complexity of inexact SQA methods has been analyzed recently. In this paper, we present an algorithmic framework of inexact SQA methods with four types of line searches, and analyze its global complexity under milder assumptions. First, we show its well-definedness and some decreasing properties. Second, under the quadratic growth condition and a uniform positive lower bound condition on stepsizes, we show that the function value sequence and the iterate sequence are linearly convergent. Moreover, we obtain a o(1/k) complexity without the quadratic growth condition, improving existing $${\mathcal {O}}(1/k)$$ O ( 1 / k ) complexity results. At last, we show that a local gradient-Lipschitz-continuity condition could guarantee a uniform positive lower bound for the stepsizes.

Suggested Citation

  • Wei Peng & Hui Zhang & Xiaoya Zhang & Lizhi Cheng, 2020. "Global complexity analysis of inexact successive quadratic approximation methods for regularized optimization under mild assumptions," Journal of Global Optimization, Springer, vol. 78(1), pages 69-89, September.
    • Handle: RePEc:spr:jglopt:v:78:y:2020:i:1:d:10.1007_s10898-020-00892-1
    DOI: 10.1007/s10898-020-00892-1
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    References listed on IDEAS

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    1. Adrien B. Taylor & Julien M. Hendrickx & François Glineur, 2018. "Exact Worst-Case Convergence Rates of the Proximal Gradient Method for Composite Convex Minimization," Journal of Optimization Theory and Applications, Springer, vol. 178(2), pages 455-476, August.
    2. Ion Necoara & Yurii Nesterov & François Glineur, 2019. "Linear convergence of first order methods for non-strongly convex optimization," LIDAM Reprints CORE 3000, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. P. Tseng & S. Yun, 2009. "Block-Coordinate Gradient Descent Method for Linearly Constrained Nonsmooth Separable Optimization," Journal of Optimization Theory and Applications, Springer, vol. 140(3), pages 513-535, March.
    4. Heinz H. Bauschke & Jérôme Bolte & Marc Teboulle, 2017. "A Descent Lemma Beyond Lipschitz Gradient Continuity: First-Order Methods Revisited and Applications," Mathematics of Operations Research, INFORMS, vol. 42(2), pages 330-348, May.
    5. Ching-pei Lee & Stephen J. Wright, 2019. "Inexact Successive quadratic approximation for regularized optimization," Computational Optimization and Applications, Springer, vol. 72(3), pages 641-674, April.
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    Cited by:

    1. Tianxiang Liu & Akiko Takeda, 2022. "An inexact successive quadratic approximation method for a class of difference-of-convex optimization problems," Computational Optimization and Applications, Springer, vol. 82(1), pages 141-173, May.

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