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A New Nonsmooth Trust Region Algorithm for Locally Lipschitz Unconstrained Optimization Problems

Author

Listed:
  • Z. Akbari

    (K.N. Toosi University of Technology)

  • R. Yousefpour

    (University of Mazandaran)

  • M. Reza Peyghami

    (K.N. Toosi University of Technology
    Scientific Computations in Optimization and Systems Engineering (SCOPE), K.N. Toosi University of Technology)

Abstract

In this paper, a new nonsmooth trust region algorithm is proposed for solving unconstrained minimization problems with locally Lipschitz objective functions. At first, by using an approximation of the steepest descent direction, a local model is presented for locally Lipschitz functions. More precisely, in the quadratic model of classical trust region methods, the gradient vector is replaced by an approximation of the steepest descent direction. We then apply one of the efficient approaches of classical trust region methods in order to solve the obtained model. Using the BFGS updating formula for the Hessian approximation of the model, we show that the proposed algorithm is convergent under some mild and standard conditions on the objective function. Finally, the presented algorithm is implemented in the MATLAB environment and applied on some nonsmooth test problems.

Suggested Citation

  • Z. Akbari & R. Yousefpour & M. Reza Peyghami, 2015. "A New Nonsmooth Trust Region Algorithm for Locally Lipschitz Unconstrained Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 164(3), pages 733-754, March.
    • Handle: RePEc:spr:joptap:v:164:y:2015:i:3:d:10.1007_s10957-014-0534-6
    DOI: 10.1007/s10957-014-0534-6
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    References listed on IDEAS

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    1. E. Polak & J. O. Royset, 2003. "Algorithms for Finite and Semi-Infinite Min–Max–Min Problems Using Adaptive Smoothing Techniques," Journal of Optimization Theory and Applications, Springer, vol. 119(3), pages 421-457, December.
    2. Jianzhong Zhang & Nae-Heon Kim & L. Lasdon, 1985. "An Improved Successive Linear Programming Algorithm," Management Science, INFORMS, vol. 31(10), pages 1312-1331, October.
    3. 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.
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    Cited by:

    1. Zengru Cui & Gonglin Yuan & Zhou Sheng & Wenjie Liu & Xiaoliang Wang & Xiabin Duan, 2015. "A Modified BFGS Formula Using a Trust Region Model for Nonsmooth Convex Minimizations," PLOS ONE, Public Library of Science, vol. 10(10), pages 1-15, October.
    2. Zhou Sheng & Gonglin Yuan, 2018. "An effective adaptive trust region algorithm for nonsmooth minimization," Computational Optimization and Applications, Springer, vol. 71(1), pages 251-271, September.

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