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A semismooth Newton method for tensor eigenvalue complementarity problem

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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.

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Acknowledgments

The authors are very grateful to the two anonymous referees for their valuable suggestions and constructive comments, which have considerably improved the presentation of the paper. We are also thankful to Dr. Ziyan Luo for her helpful comments. Liqun Qi work was supported by the Hong Kong Research Grant Council (Grant No. PolyU 502111, 501212, 501913 and 15302114).

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Correspondence to Zhongming Chen.

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Zhongming Chen work was done when he was visiting the Hong Kong Polytechnic University.

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Chen, Z., Qi, L. A semismooth Newton method for tensor eigenvalue complementarity problem. Comput Optim Appl 65, 109–126 (2016). https://doi.org/10.1007/s10589-016-9838-9

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