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Adaptive smoothing algorithms for nonsmooth composite convex minimization

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  • Quoc Tran-Dinh

    (University of North Carolina at Chapel Hill (UNC))

Abstract

We propose an adaptive smoothing algorithm based on Nesterov’s smoothing technique in Nesterov (Math Prog 103(1):127–152, 2005) for solving “fully” nonsmooth composite convex optimization problems. Our method combines both Nesterov’s accelerated proximal gradient scheme and a new homotopy strategy for smoothness parameter. By an appropriate choice of smoothing functions, we develop a new algorithm that has the $$\mathcal {O}\left( \frac{1}{\varepsilon }\right) $$ O 1 ε -worst-case iteration-complexity while preserves the same complexity-per-iteration as in Nesterov’s method and allows one to automatically update the smoothness parameter at each iteration. Then, we customize our algorithm to solve four special cases that cover various applications. We also specify our algorithm to solve constrained convex optimization problems and show its convergence guarantee on a primal sequence of iterates. We demonstrate our algorithm through three numerical examples and compare it with other related algorithms.

Suggested Citation

  • Quoc Tran-Dinh, 2017. "Adaptive smoothing algorithms for nonsmooth composite convex minimization," Computational Optimization and Applications, Springer, vol. 66(3), pages 425-451, April.
    • Handle: RePEc:spr:coopap:v:66:y:2017:i:3:d:10.1007_s10589-016-9873-6
    DOI: 10.1007/s10589-016-9873-6
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    References listed on IDEAS

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    1. NESTEROV, Yurii, 2007. "Smoothing technique and its applications in semidefinite optimization," LIDAM Reprints CORE 1951, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. A. Belloni & V. Chernozhukov & L. Wang, 2011. "Square-root lasso: pivotal recovery of sparse signals via conic programming," Biometrika, Biometrika Trust, vol. 98(4), pages 791-806.
    3. Radu Boţ & Christopher Hendrich, 2013. "A double smoothing technique for solving unconstrained nondifferentiable convex optimization problems," Computational Optimization and Applications, Springer, vol. 54(2), pages 239-262, March.
    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. DEVOLDER, Olivier & GLINEUR, François & NESTEROV, Yurii, 2012. "Double smoothing technique for large-scale linearly constrained convex optimization," LIDAM Reprints CORE 2423, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    6. Quoc Tran Dinh & Carlo Savorgnan & Moritz Diehl, 2013. "Combining Lagrangian decomposition and excessive gap smoothing technique for solving large-scale separable convex optimization problems," Computational Optimization and Applications, Springer, vol. 55(1), pages 75-111, May.
    7. Patrick L. Combettes & Jean-Christophe Pesquet, 2011. "Proximal Splitting Methods in Signal Processing," Springer Optimization and Its Applications, in: Heinz H. Bauschke & Regina S. Burachik & Patrick L. Combettes & Veit Elser & D. Russell Luke & Henry (ed.), Fixed-Point Algorithms for Inverse Problems in Science and Engineering, chapter 0, pages 185-212, Springer.
    8. 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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    Cited by:

    1. Fan Wu & Wei Bian, 2023. "Smoothing Accelerated Proximal Gradient Method with Fast Convergence Rate for Nonsmooth Convex Optimization Beyond Differentiability," Journal of Optimization Theory and Applications, Springer, vol. 197(2), pages 539-572, May.

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