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Efficiency of higher-order algorithms for minimizing composite functions

Author

Listed:
  • Yassine Nabou

    (National University of Science and Technology Politehnica Bucharest)

  • Ion Necoara

    (National University of Science and Technology Politehnica Bucharest
    Gheorghe Mihoc-Caius Iacob Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy)

Abstract

Composite minimization involves a collection of functions which are aggregated in a nonsmooth manner. It covers, as a particular case, smooth approximation of minimax games, minimization of max-type functions, and simple composite minimization problems, where the objective function has a nonsmooth component. We design a higher-order majorization algorithmic framework for fully composite problems (possibly nonconvex). Our framework replaces each component with a higher-order surrogate such that the corresponding error function has a higher-order Lipschitz continuous derivative. We present convergence guarantees for our method for composite optimization problems with (non)convex and (non)smooth objective function. In particular, we prove stationary point convergence guarantees for general nonconvex (possibly nonsmooth) problems and under Kurdyka–Lojasiewicz (KL) property of the objective function we derive improved rates depending on the KL parameter. For convex (possibly nonsmooth) problems we also provide sublinear convergence rates.

Suggested Citation

  • Yassine Nabou & Ion Necoara, 2024. "Efficiency of higher-order algorithms for minimizing composite functions," Computational Optimization and Applications, Springer, vol. 87(2), pages 441-473, March.
    • Handle: RePEc:spr:coopap:v:87:y:2024:i:2:d:10.1007_s10589-023-00533-9
    DOI: 10.1007/s10589-023-00533-9
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    References listed on IDEAS

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    1. NESTEROV, Yurii, 2013. "Gradient methods for minimizing composite functions," LIDAM Reprints CORE 2510, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Nicholas I. M. Gould & Tyrone Rees & Jennifer A. Scott, 2019. "Convergence and evaluation-complexity analysis of a regularized tensor-Newton method for solving nonlinear least-squares problems," Computational Optimization and Applications, Springer, vol. 73(1), pages 1-35, May.
    3. 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).
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