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A convergence analysis of the method of codifferential descent

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  • M. V. Dolgopolik

    (Saint Petersburg State University
    Russian Academy of Sciences)

Abstract

This paper is devoted to a detailed convergence analysis of the method of codifferential descent (MCD) developed by professor V.F. Demyanov for solving a large class of nonsmooth nonconvex optimization problems. We propose a generalization of the MCD that is more suitable for applications than the original method, and that utilizes only a part of a codifferential on every iteration, which allows one to reduce the overall complexity of the method. With the use of some general results on uniformly codifferentiable functions obtained in this paper, we prove the global convergence of the generalized MCD in the infinite dimensional case. Also, we propose and analyse a quadratic regularization of the MCD, which is the first general method for minimizing a codifferentiable function over a convex set. Apart from convergence analysis, we also discuss the robustness of the MCD with respect to computational errors, possible step size rules, and a choice of parameters of the algorithm. In the end of the paper we estimate the rate of convergence of the MCD for a class of nonsmooth nonconvex functions that arise, in particular, in cluster analysis. We prove that under some general assumptions the method converges with linear rate, and it convergence quadratically, provided a certain first order sufficient optimality condition holds true.

Suggested Citation

  • M. V. Dolgopolik, 2018. "A convergence analysis of the method of codifferential descent," Computational Optimization and Applications, Springer, vol. 71(3), pages 879-913, December.
    • Handle: RePEc:spr:coopap:v:71:y:2018:i:3:d:10.1007_s10589-018-0024-0
    DOI: 10.1007/s10589-018-0024-0
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    References listed on IDEAS

    as
    1. W. Hare & J. Nutini, 2013. "A derivative-free approximate gradient sampling algorithm for finite minimax problems," Computational Optimization and Applications, Springer, vol. 56(1), pages 1-38, September.
    2. A. Bagirov & J. Ugon, 2011. "Codifferential method for minimizing nonsmooth DC functions," Journal of Global Optimization, Springer, vol. 50(1), pages 3-22, May.
    3. A. M. Bagirov & B. Karasözen & M. Sezer, 2008. "Discrete Gradient Method: Derivative-Free Method for Nonsmooth Optimization," Journal of Optimization Theory and Applications, Springer, vol. 137(2), pages 317-334, May.
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    Cited by:

    1. M. V. Dolgopolik, 2020. "New global optimality conditions for nonsmooth DC optimization problems," Journal of Global Optimization, Springer, vol. 76(1), pages 25-55, January.

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