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A Modified BFGS Formula Using a Trust Region Model for Nonsmooth Convex Minimizations

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Listed:
  • Zengru Cui
  • Gonglin Yuan
  • Zhou Sheng
  • Wenjie Liu
  • Xiaoliang Wang
  • Xiabin Duan

Abstract

This paper proposes a modified BFGS formula using a trust region model for solving nonsmooth convex minimizations by using the Moreau-Yosida regularization (smoothing) approach and a new secant equation with a BFGS update formula. Our algorithm uses the function value information and gradient value information to compute the Hessian. The Hessian matrix is updated by the BFGS formula rather than using second-order information of the function, thus decreasing the workload and time involved in the computation. Under suitable conditions, the algorithm converges globally to an optimal solution. Numerical results show that this algorithm can successfully solve nonsmooth unconstrained convex problems.

Suggested Citation

  • 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.
    • Handle: RePEc:plo:pone00:0140606
    DOI: 10.1371/journal.pone.0140606
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    References listed on IDEAS

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    1. Sha Lu & Zengxin Wei & Lue Li, 2012. "A trust region algorithm with adaptive cubic regularization methods for nonsmooth convex minimization," Computational Optimization and Applications, Springer, vol. 51(2), pages 551-573, March.
    2. Correa Romar, 2014. "Mathematical Foci," Mathematical Economics Letters, De Gruyter, vol. 2(1-2), pages 5-11, August.
    3. 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.
    4. Gonglin Yuan & Zengxin Wei & Zhongxing Wang, 2013. "Gradient trust region algorithm with limited memory BFGS update for nonsmooth convex minimization," Computational Optimization and Applications, Springer, vol. 54(1), pages 45-64, January.
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    Cited by:

    1. Tsegay Giday Woldu & Haibin Zhang & Xin Zhang & Yemane Hailu Fissuh, 2020. "A Modified Nonlinear Conjugate Gradient Algorithm for Large-Scale Nonsmooth Convex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 185(1), pages 223-238, April.
    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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