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A semismooth Newton method for tensor eigenvalue complementarity problem

Author

Listed:
  • Zhongming Chen

    () (Nankai University)

  • Liqun Qi

    () (The Hong Kong Polytechnic University)

Abstract

Abstract In this paper, we consider the tensor eigenvalue complementarity problem which is closely related to the optimality conditions for polynomial optimization, as well as a class of differential inclusions with nonconvex processes. By introducing an NCP-function, we reformulate the tensor eigenvalue complementarity problem as a system of nonlinear equations. We show that this function is strongly semismooth but not differentiable, in which case the classical smooth methods cannot apply. Furthermore, we propose a damped semismooth Newton method for tensor eigenvalue complementarity problem. A new procedure to evaluate an element of the generalized Jacobian is given, which turns out to be an element of the B-subdifferential under mild assumptions. As a result, the convergence of the damped semismooth Newton method is guaranteed by existing results. The numerical experiments also show that our method is efficient and promising.

Suggested Citation

  • Zhongming Chen & Liqun Qi, 2016. "A semismooth Newton method for tensor eigenvalue complementarity problem," Computational Optimization and Applications, Springer, vol. 65(1), pages 109-126, September.
  • Handle: RePEc:spr:coopap:v:65:y:2016:i:1:d:10.1007_s10589-016-9838-9
    DOI: 10.1007/s10589-016-9838-9
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    References listed on IDEAS

    as
    1. Chen Ling & Hongjin He & Liqun Qi, 2016. "On the cone eigenvalue complementarity problem for higher-order tensors," Computational Optimization and Applications, Springer, vol. 63(1), pages 143-168, January.
    2. Chen Ling & Hongjin He & Liqun Qi, 2016. "On the cone eigenvalue complementarity problem for higher-order tensors," Computational Optimization and Applications, Springer, vol. 63(1), pages 143-168, January.
    3. Samir Adly & Hadia Rammal, 2013. "A new method for solving Pareto eigenvalue complementarity problems," Computational Optimization and Applications, Springer, vol. 55(3), pages 703-731, July.
    4. Yisheng Song & Liqun Qi, 2016. "Eigenvalue analysis of constrained minimization problem for homogeneous polynomial," Journal of Global Optimization, Springer, vol. 64(3), pages 563-575, March.
    5. Qin Ni & Liqun Qi, 2015. "A quadratically convergent algorithm for finding the largest eigenvalue of a nonnegative homogeneous polynomial map," Journal of Global Optimization, Springer, vol. 61(4), pages 627-641, April.
    6. Yisheng Song & Liqun Qi, 2016. "Eigenvalue analysis of constrained minimization problem for homogeneous polynomial," Journal of Global Optimization, Springer, vol. 64(3), pages 563-575, March.
    7. A. Pinto da Costa & A. Seeger, 2010. "Cone-constrained eigenvalue problems: theory and algorithms," Computational Optimization and Applications, Springer, vol. 45(1), pages 25-57, January.
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