Tensor Eigenvalues and Their Applications by Liqun Qi Haibin Chen & Yannan Chen

Author:Liqun Qi, Haibin Chen & Yannan Chen
Language: eng
Format: epub
Publisher: Springer Singapore, Singapore

5.6 Notes

The tensor eigenvalue complementarity problem is a companion problem of the tensor complementarity problem [126]. In this chapter, we study the TEiCP from theory point of view to calculation methods. It should be noted that the topic on this direction still need to study in the future since the convergence analysis of the proposed algorithm and designing algorithms for nonsymmetric cases need further study.

Section 5.1 The contents in this section was first given by Ling, He, and Qi in [179]. For the sake of completeness, we give a detailed proof for Theorem 5.5.

Section 5.2 We study properties of Pareto H-eigenvalues and Pareto Z-eigenvalues in this section. It was originally defined by Song and Qi in [249]. Related definitions about H-eigenvalue, H-eigenvalue, Z-eigenvalue and Z-eigenvalue can check the book [228].

Section 5.3 In this section, the damped semi-smooth Newton method was first presented by Chen and Qi in [56]. Proposition 5.10 was originally proved by Clarke in [60], and Proposition 5.11 was proved by Chen, Chen, and Kanzow in [40]. The scaling-and-projection algorithm was first introduced by Costa and Seeger in [73], for solving matrix cone constrained eigenvalue problems. Motivated by this, Ling He and Qi gave a new form of the algorithm for solving the tensor case [179].

Section 5.4 The main results of this section was originally given by Ling, He and Qi in [180].

Section 5.5 Fan, Nie, and Zhou gave the semidefinite relaxation method in [93]. It is a polynomial method that can be solved by Lasserre type semidefinite relaxations, which was first given by Lasserre in [162, 163]. Nie et al. developed this method in [200, 201, 203–207]. By the way, it should be noted that the Lasserre type semidefinite relaxations (5.103) and (5.107) can be solved by the software GloptiPoly 3 [122] and SeDuMi [255].

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