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Acceleration techniques for level bundle methods in weakly smooth convex constrained optimization

Author

Listed:
  • Yunmei Chen

    (University of Florida)

  • Xiaojing Ye

    (George State University)

  • Wei Zhang

    (University of Florida)

Abstract

We develop a unified level-bundle method, called accelerated constrained level-bundle (ACLB) algorithm, for solving constrained convex optimization problems. where the objective and constraint functions can be nonsmooth, weakly smooth, and/or smooth. ACLB employs Nesterov’s accelerated gradient technique, and hence retains the iteration complexity as that of existing bundle-type methods if the objective or one of the constraint functions is nonsmooth. More importantly, ACLB can significantly reduce iteration complexity when the objective and all constraints are (weakly) smooth. In addition, if the objective contains a nonsmooth component which can be written as a specific form of maximum, we show that the iteration complexity of this component can be much lower than that for general nonsmooth objective function. Numerical results demonstrate the effectiveness of the proposed algorithm.

Suggested Citation

  • Yunmei Chen & Xiaojing Ye & Wei Zhang, 2020. "Acceleration techniques for level bundle methods in weakly smooth convex constrained optimization," Computational Optimization and Applications, Springer, vol. 77(2), pages 411-432, November.
    • Handle: RePEc:spr:coopap:v:77:y:2020:i:2:d:10.1007_s10589-020-00208-9
    DOI: 10.1007/s10589-020-00208-9
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    References listed on IDEAS

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    1. J. Bello Cruz & W. Oliveira, 2014. "Level bundle-like algorithms for convex optimization," Journal of Global Optimization, Springer, vol. 59(4), pages 787-809, August.
    2. Yunmei Chen & Guanghui Lan & Yuyuan Ouyang & Wei Zhang, 2019. "Fast bundle-level methods for unconstrained and ball-constrained convex optimization," Computational Optimization and Applications, Springer, vol. 73(1), pages 159-199, May.
    3. Robert Tibshirani & Michael Saunders & Saharon Rosset & Ji Zhu & Keith Knight, 2005. "Sparsity and smoothness via the fused lasso," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(1), pages 91-108, February.
    4. Wim Ackooij & Welington Oliveira, 2014. "Level bundle methods for constrained convex optimization with various oracles," Computational Optimization and Applications, Springer, vol. 57(3), pages 555-597, April.
    5. Lemaréchal, C. & Nemirovskii, A. & Nesterov, Y., 1995. "New variants of bundle methods," LIDAM Reprints CORE 1166, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    6. NESTEROV, Yu., 2005. "Smooth minimization of non-smooth functions," LIDAM Reprints CORE 1819, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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