\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Proximal iterative Gaussian smoothing algorithm for a class of nonsmooth convex minimization problems

Abstract Related Papers Cited by
  • In this paper, we consider the problem of minimizing a convex objective which is the sum of three parts: a smooth part, a simple non-smooth Lipschitz part, and a simple non-smooth non-Lipschitz part. A novel optimization algorithm is proposed for solving this problem. By making use of the Gaussian smoothing function of the functions occurring in the objective, we smooth the second part to a convex and differentiable function with Lipschitz continuous gradient by using both variable and constant smoothing parameters. The resulting problem is solved via an accelerated proximal-gradient method and this allows us to recover approximately the optimal solutions to the initial optimization problem with a rate of convergence of order $O(\frac{\ln k}{k})$ for variable smoothing and of order $O(\frac{1}{k})$ for constant smoothing.
    Mathematics Subject Classification: Primary: 90C30; Secondary: 90C25.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, CMS Books in Mathematics, Springer, 2011.doi: 10.1007/978-1-4419-9467-7.

    [2]

    A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM Journal of Imaging Sciences, 2 (2009), 183-202.doi: 10.1137/080716542.

    [3]

    D. P. Bertsekas, Nonlinear Programming, Athena Scientific, Belmont, 1999.

    [4]

    E. J. Candes, X. Li, Y. Ma and J. Wright, Robust principal component analysis, Journal of the Association for Computing Machinery, 58 (2011), 219-226.doi: 10.1145/1970392.1970395.

    [5]

    J. Duchi and Y. Singer, Efficient online and batch learning using forward backward splitting, The Journal of Machine Learning Research, 10 (2009), 2899-2934.

    [6]

    Y. Nesterov, Introductory lectures on convex optimization: A basic course, Springer, 2004.doi: 10.1007/978-1-4419-8853-9.

    [7]

    Y. Nesterov, Smooth minimization of non-smooth functions, Mathematical Programming, 103 (2005b), 127-152. doi: 10.1007/s10107-004-0552-5.

    [8]

    Y. Nesterov, Random gradient-free minimization of convex functions, Core discussion paper, 2001, http://www.uclouvain.be/core.

    [9]

    R. T. Rockafellar, Convex Analysis, Princeton University Press, 1970.

    [10]

    M. J. Wainwright, P. Ravikumar and J. D. Lafferty, High-dimensional graphical model selection using l1-regularized logistic regression, In Advances in Neural Information Processing Systems,19 (2007),1465-1472.

    [11]

    M. Yuan and Y. Lin, Model selection and estimation in the Gaussian graphical model, Biometrika, 94 (2007), 19-35.doi: 10.1093/biomet/asm018.

    [12]

    C. Zalinescu, Convex Analysis in General Vector Spaces, World Scientific, 2002.doi: 10.1142/9789812777096.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(88) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return