Abstract
The main purpose of this paper was to investigate some kinds of nonlinear complementarity problems (NCP). For the structured tensors, such as symmetric positive-definite tensors and copositive tensors, we derive the existence theorems on a solution of these kinds of nonlinear complementarity problems. We prove that a unique solution of the NCP exists under the condition of diagonalizable tensors.
Highlights
Let F be a mapping from Rn into itself
Lim [10] proposed another definition of eigenvalues, eigenvectors, singular values, and singular vectors for tensors based on a constrained variational approach, much like the Rayleigh quotient for symmetric matrix eigenvalues, independently. It is well-known that A is a P matrix ([1]), if and only if the linear complementarity problem, find z ∈ Rn such that z ≥ 0, q + Az ≥ 0, and z⊤(q + Az) = 0
(a) If A is diagonalizable and positive definite, the nonlinear complementarity problems (NCP)(q, A) has a unique solution, (b) If A is positive definite, the NCP(q, A) has a nonempty, compact solution set, (c) If A is strictly copositive with respect to Rn+, the NCP(q, A) has a nonempty, compact solution set
Summary
The nonlinear complementarity problem, denoted by NCP(F ), is to find a vector x∗ ∈ Rn+ such that. Lim [10] proposed another definition of eigenvalues, eigenvectors, singular values, and singular vectors for tensors based on a constrained variational approach, much like the Rayleigh quotient for symmetric matrix eigenvalues, independently. It is well-known that A is a P matrix ([1]), if and only if the linear complementarity problem, find z ∈ Rn such that z ≥ 0, q + Az ≥ 0, and z⊤(q + Az) = 0.
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