A smoothing Newton method preserving nonnegativity for solving tensor complementarity problems with $ P_0 $ mappings

Abstract

In this paper, we prove that the tensor complementarity problem with the \begin{document}$ P_0 $\end{document} mapping on the \begin{document}$ n $\end{document}-dimensional nonnegative orthant is solvable and the solution set is nonempty and compact under mild assumptions. Since the involved homogeneous polynomial is a \begin{document}$ P_0 $\end{document} mapping on the \begin{document}$ n $\end{document}-dimensional nonnegative orthant, the existing smoothing Newton methods are not directly used to solve this problem. So, we propose a smoothing Newton method preserving nonnegativity via a new one-dimensional line search rule for solving such problem. The global convergence is established and preliminary numerical results illustrate that the proposed algorithm is efficient and very promising.