Abstract:We focus on establishing an algorithm to solve the tensor eigenvalue complementarity problem (TEiCP), and we have two contributions in this paper. First, a smoothing Newton-type algorithm is proposed for the TEiCP based on the CHKS smoothing function. Its global convergence is established under some mild conditions. Numerical experiments are reported to show that the proposed algorithm is efficient and could detect more solutions than some existing methods. Second, we apply the proposed algorithm to solve the eigenvalue problem of nonnegative tensors. We analyze the relationship between the TEiCP and the H-eigenpair and Z-eigenpair problems of an irreducible nonnegative tensor. We show that the TEiCP with an irreducible nonnegative tensor and unit tensor has a unique solution, which is just the unique positive H-eigenpair of the irreducible nonnegative tensor. We also show that the solution set of the TEiCP with an irreducible nonnegative tensor and identity tensor is nonempty and its solutions are positive. Moreover, we can obtain positive Z-eigenpairs of the irreducible nonnegative tensor from these solutions. Finally, we also apply the proposed algorithm to find the unique positive H-eigenpair and a positive Z-eigenpair of an irreducible nonnegative tensor; the numerical results indicate its efficiency and promising performance.

---

中文翻译：

---

张量特征值互补问题的牛顿型算法及其应用

摘要：我们致力于建立一种解决张量特征值互补性问题的算法，本文有两个贡献。首先，提出了一种基于CHKS平滑函数的TEiCP平滑牛顿型算法。它的全球趋同是在某些温和条件下建立的。数值实验表明，与现有方法相比，该算法是有效的，可以检测出更多的解决方案。其次，我们将提出的算法用于解决非负张量的特征值问题。我们分析了TEiCP与不可约非负张量的H本征对和Z本征对之间的关​​系。我们证明，具有不可约的非负张量和单位张量的TEiCP具有独特的解决方案，这只是不可约非负张量的唯一正H本征对。我们还表明，具有不可约非负张量和恒等张量的TEiCP的解集是非空的，并且其解是正的。此外，我们可以从这些解中获得不可约非负张量的正Z本征对。最后，我们还应用提出的算法来找到不可约非负张量的唯一正H特征对和正Z特征对。数值结果表明其效率和良好的性能。我们还应用提出的算法来找到不可约非负张量的唯一正H特征对和正Z特征对。数值结果表明其效率和良好的性能。我们还应用提出的算法来找到不可约非负张量的唯一正H特征对和正Z特征对; 数值结果表明其效率和良好的性能。

---