“…We emphasize that the following result holds without making any additional hypotheses on A, B, C. We also mention that the proof of Proposition 2.3(b) is quite different from the proof of its counterpart for the case of K = R n + , namely Proposition 1 in [7].…”
Section: Existence Of Solutions Of Cqeicpmentioning
confidence: 88%
“…Given A, B, C ∈ R n×n , QEiCP(A, B, C) consists of finding (λ, x, w) ∈ R × R n × R n such that w = λ 2 Ax + λBx + Cx, (6) w ≥ 0, x ≥ 0, (7) x t w = 0, (8) e t x = 1, (9) where, as before, e = (1, 1, . .…”
Section: Introductionmentioning
confidence: 99%
“…As in the case of the EiCP, the normalization constraint (9) The case of the symmetric QEiCP, i.e., when A, B and C are symmetric matrices and −C is the identity matrix, has been analyzed in [13], where each instance of QEiCP with n × n matrices is related to an instance of EiCP with 2n × 2n matrices. A new approach for solving the nonsymmetric QEiCP by a similar reduction has been recently studied in [7].…”
Section: Introductionmentioning
confidence: 99%
“…In [31], the concepts of co-regularity and co-hyperbolicity of (A, B, C) were introduced, ensuring existence of solutions of CQEiCP(A, B, C). For the case of QEiCP (i.e., when K = R n + ), it has been shown in [7] that existence of solutions of QEiCP is also guaranteed when the matrix A is strictly copositive and the matrix −C is not an S 0 matrix. In order to establish this result, QEiCP is transformed into a 2n-dimensional EiCP problem by using an auxiliary vector y ∈ R n such that y = λx.…”
Section: Introductionmentioning
confidence: 99%
“…In this paper we propose a new transformation of CQEiCP into CEiCP (for a general closed, convex and pointed cone K) that differs from the one introduced in [7] by the introduction of a PD matrix E. Using this transformation, we will establish in Section 2 the existence of solutions of CQEiCP under hypotheses different from those demanded in [31].…”
“…We emphasize that the following result holds without making any additional hypotheses on A, B, C. We also mention that the proof of Proposition 2.3(b) is quite different from the proof of its counterpart for the case of K = R n + , namely Proposition 1 in [7].…”
Section: Existence Of Solutions Of Cqeicpmentioning
confidence: 88%
“…Given A, B, C ∈ R n×n , QEiCP(A, B, C) consists of finding (λ, x, w) ∈ R × R n × R n such that w = λ 2 Ax + λBx + Cx, (6) w ≥ 0, x ≥ 0, (7) x t w = 0, (8) e t x = 1, (9) where, as before, e = (1, 1, . .…”
Section: Introductionmentioning
confidence: 99%
“…As in the case of the EiCP, the normalization constraint (9) The case of the symmetric QEiCP, i.e., when A, B and C are symmetric matrices and −C is the identity matrix, has been analyzed in [13], where each instance of QEiCP with n × n matrices is related to an instance of EiCP with 2n × 2n matrices. A new approach for solving the nonsymmetric QEiCP by a similar reduction has been recently studied in [7].…”
Section: Introductionmentioning
confidence: 99%
“…In [31], the concepts of co-regularity and co-hyperbolicity of (A, B, C) were introduced, ensuring existence of solutions of CQEiCP(A, B, C). For the case of QEiCP (i.e., when K = R n + ), it has been shown in [7] that existence of solutions of QEiCP is also guaranteed when the matrix A is strictly copositive and the matrix −C is not an S 0 matrix. In order to establish this result, QEiCP is transformed into a 2n-dimensional EiCP problem by using an auxiliary vector y ∈ R n such that y = λx.…”
Section: Introductionmentioning
confidence: 99%
“…In this paper we propose a new transformation of CQEiCP into CEiCP (for a general closed, convex and pointed cone K) that differs from the one introduced in [7] by the introduction of a PD matrix E. Using this transformation, we will establish in Section 2 the existence of solutions of CQEiCP under hypotheses different from those demanded in [31].…”
In this paper, we introduce a unified framework of Tensor Higher-Degree Eigenvalue Complementarity Problem (THDEiCP), which goes beyond the framework of the typical Quadratic Eigenvalue Complementarity Problem (QEiCP) for matrices. First, we study some topological properties of higher-degree cone eigenvalues of tensors. Based upon the symmetry assumptions on the underlying tensors, we then reformulate THDEiCP as a weakly coupled homogeneous polynomial optimization problem, which might be greatly helpful for designing implementable algorithms to solve the problem under consideration numerically. As more general theoretical results, we present the results concerning existence of solutions of THDEiCP without symmetry conditions. Finally, we propose an easily implementable algorithm to solve THDEiCP, and report some computational results.Keywords Tensor · Higher-degree cone eigenvalue · Eigenvalue complementarity problem · Polynomial optimization problem · Augmented Lagrangian method ·
Alternating direction method of multipliersMathematics Subject Classification (2010) 15A18 · 15A69 · 65K15 · 90C30 · 90C33
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