Article Contents
Article Contents

# Modified spectral PRP conjugate gradient method for solving tensor eigenvalue complementarity problems

• * Corresponding author: ShouQiang Du
• Tensor eigenvalue complementary problems, as a special class of complementary problems, are the generalization of matrix eigenvalue complementary problems in higher-order. In recent years, tensor eigenvalue complementarity problems have been studied extensively. The research fields of tensor eigenvalue complementarity problems mainly focus on analysis of the theory and algorithms. In this paper, we investigate the solution method for four kinds of tensor eigenvalue complementarity problems with different structures. By utilizing an equivalence relation to unconstrained optimization problems, we propose a modified spectral PRP conjugate gradient method to solve the tensor eigenvalue complementarity problems. Under mild conditions, the global convergence of the given method is also established. Finally, we give related numerical experiments and numerical results compared with inexact Levenberg-Marquardt method, numerical results show the efficiency of the proposed method and also verify our theoretical results.

Mathematics Subject Classification: Primary: 90C30, 65K05; Secondary: 15A18.

 Citation:

• Figure 1.  Numerical results of Example 4.1 with random initial points

Figure 2.  Numerical results of Example 4.5 with different initial points

Table 1.  The numerical results of Example 4.1

 Eigvalue Eigvector No 1.9406 $(0.4982, 0.5018)^T$ 24 2.0469 $(0.8817, 0.1183)^T$ 5 2.6308 $(0.0000, 1.0000)^T$ 1

Table 2.  The numerical results of Example 4.2

 Alg. Eigvalue Eigvector No. K Algorithm 3.1 0.3024 $(0.5479, 1.0000, 0.4521)^T$ 3 69 Algorithm 3.1 0.2356 $(0.5037, 0.4963, 0.0000)^T$ 2 56 Algorithm 3.1 0.2089 $(0.0000, 0.4911, 0.5089)^T$ 1 41 Algorithm 3.1 0.0771 $(0.3328, 0.3372, 0.3301)^T$ 3 183 Algorithm 3.1 $failure$ $-$ 1 - ILMM 0.2089 $(0.0000, 0.4911, 0.5089)^T$ 2 26 ILMM 0.2356 $(0.5037, 0.4963, 0.0000)^T$ 2 41 ILMM 0.3024 $(0.5479, 1.0000, 0.4521)^T$ 1 45 ILMM 0.9807 $(0.0000, 1.0000, 0.0000)^T$ 1 119 ILMM $failure$ - 4 -

Table 3.  The numerical results of Example 4.3

 Eigvalue Eigvector No 0.8867 $(1.0000, 0.0000)^T$ 2 0.9533 $(0.0000, 1.0000)^T$ 4 0.6 $(0.6, 0.4)^T$ 14

Table 4.  The numerical results of Example 4.4

 Eigvalue Eigvector No 2.9167 $(0.0232, 0.0000, 0.9967)^T$ 3 2.9865 $(0.0712, 0.0252, 0.9971)^T$ 2 0.3 $(0.1, 0, 0.1)^T$ 1 failure 4
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