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A generalized Newton method for absolute value equations associated with circular cones

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  • Miao, Xin-He
  • Yang, Jiantao
  • Hu, Shenglong

Abstract

In this paper, we study a class of absolute value equations associated with circular cone (CCAVE for short), which is a generalization of the absolute value equations discussed recently in the literature, analogously to the fact that circular cone is the very generalization of the second-order cone. We show that the CCAVE is equivalent to a class of circular cone linear complementarity problems, and hence generalize the well-known equivalence between absolute value equations and linear complementarity problems. Useful properties of the generalized differential of the absolute value function over the circular cone are investigated, which helps to propose a generalized Newton method for solving the CCAVE. The convergence of this method is established under mild conditions, as well as the efficiency of which is illustrated by some preliminary numerical results.

Suggested Citation

  • Miao, Xin-He & Yang, Jiantao & Hu, Shenglong, 2015. "A generalized Newton method for absolute value equations associated with circular cones," Applied Mathematics and Computation, Elsevier, vol. 269(C), pages 155-168.
    • Handle: RePEc:eee:apmaco:v:269:y:2015:i:c:p:155-168
    DOI: 10.1016/j.amc.2015.07.064
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    References listed on IDEAS

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    1. Louis Caccetta & Biao Qu & Guanglu Zhou, 2011. "A globally and quadratically convergent method for absolute value equations," Computational Optimization and Applications, Springer, vol. 48(1), pages 45-58, January.
    2. Oleg Prokopyev, 2009. "On equivalent reformulations for absolute value equations," Computational Optimization and Applications, Springer, vol. 44(3), pages 363-372, December.
    3. Zheng-Hai Huang & Tie Ni, 2010. "Smoothing algorithms for complementarity problems over symmetric cones," Computational Optimization and Applications, Springer, vol. 45(3), pages 557-579, April.
    4. C. Zhang & Q. J. Wei, 2009. "Global and Finite Convergence of a Generalized Newton Method for Absolute Value Equations," Journal of Optimization Theory and Applications, Springer, vol. 143(2), pages 391-403, November.
    5. Olvi L. Mangasarian, 2014. "Absolute Value Equation Solution Via Linear Programming," Journal of Optimization Theory and Applications, Springer, vol. 161(3), pages 870-876, June.
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    Cited by:

    1. Jingyong Tang & Jinchuan Zhou, 2020. "Smoothing inexact Newton method based on a new derivative-free nonmonotone line search for the NCP over circular cones," Annals of Operations Research, Springer, vol. 295(2), pages 787-808, December.
    2. Shota Yamanaka & Nobuo Yamashita, 2018. "Duality of nonconvex optimization with positively homogeneous functions," Computational Optimization and Applications, Springer, vol. 71(2), pages 435-456, November.

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