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A conjugate gradient sampling method for nonsmooth optimization

Author

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  • N. Mahdavi-Amiri

    (Sharif University of Technology)

  • M. Shaeiri

    (Sharif University of Technology)

Abstract

We present an algorithm for minimizing locally Lipschitz functions being continuously differentiable in an open dense subset of $$\mathbb {R}^n$$Rn. The function may be nonsmooth and/or nonconvex. The method makes use of a gradient sampling method along with a conjugate gradient scheme. To find search directions, we make use of a sequence of positive definite approximate Hessians based on conjugate gradient matrices. The algorithm benefits from a restart procedure to improve upon poor search directions or to make sure that the approximate Hessians remain bounded. The global convergence of the algorithm is established. An implementation of the algorithm is executed on a collection of well-known test problems. Comparative numerical results clearly show outperformance of the algorithm over some recent well-known nonsmooth algorithms using the Dolan–Moré performance profiles.

Suggested Citation

  • N. Mahdavi-Amiri & M. Shaeiri, 2020. "A conjugate gradient sampling method for nonsmooth optimization," 4OR, Springer, vol. 18(1), pages 73-90, March.
    • Handle: RePEc:spr:aqjoor:v:18:y:2020:i:1:d:10.1007_s10288-019-00404-2
    DOI: 10.1007/s10288-019-00404-2
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    References listed on IDEAS

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    1. Nezam Mahdavi-Amiri & Rohollah Yousefpour, 2012. "An Effective Nonsmooth Optimization Algorithm for Locally Lipschitz Functions," Journal of Optimization Theory and Applications, Springer, vol. 155(1), pages 180-195, October.
    2. David F. Shanno, 1978. "Conjugate Gradient Methods with Inexact Searches," Mathematics of Operations Research, INFORMS, vol. 3(3), pages 244-256, August.
    3. Andrei, Neculai, 2010. "Accelerated scaled memoryless BFGS preconditioned conjugate gradient algorithm for unconstrained optimization," European Journal of Operational Research, Elsevier, vol. 204(3), pages 410-420, August.
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