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Global Uniqueness and Solvability for Tensor Complementarity Problems
Xue-Li Bai, Zheng-Hai Huang, Yong Wang
TL;DR
The paper examines whether P tensors characterize global uniqueness and solvability for tensor complementarity problems. It uses counterexamples to answer this question negatively, proves nonemptiness and compactness of solution sets, and introduces strong P tensors whose corresponding TCPs have the GUS-property.
Problem
The paper investigates whether TCP(q, A) has the GUS-property if and only if A is a P tensor, extending the known LCP characterization.
Method
The paper constructs two counterexamples, analyzes solution sets for P tensors, and introduces the strong P tensor class.
Results
P tensors do not characterize the GUS-property: the paper gives counterexamples, proves P-tensor TCP solution sets are nonempty and compact, and shows strong P tensors guarantee GUS.
Takeaways & Limitations
Strong P tensors provide a sufficient class for the GUS-property and form a proper subset of the P tensors.
Takeaways & Limitations
Further properties of strong P tensors remain to be studied using known methods and results for P-functions.
Abstract
from arXiv · showhide
Recently, the tensor complementarity problem (TCP for short) has been investigated in the literature. An important question involving the property of global uniqueness and solvability (GUS-property) for a class of TCPs was proposed by Song and Qi in their paper "Properties of Some Classes of Structured Tensors". In the present paper, we give an answer to this question by constructing two counter-examples. We also show that the solution set of this class of TCPs is nonempty and compact. In particular, we introduce a class of related structured tensors, and show that the corresponding TCP has the GUS-property.
1 Introduction
The paper addresses whether the P-tensor characterization of global uniqueness and solvability extends from linear complementarity problems to tensor complementarity problems. It answers negatively with counterexamples and establishes compactness and a sufficient tensor class for GUS.
- Research question: The paper asks whether TCP(q, A) has the GUS-property if and only if A is a P tensor.This question was proposed by Song and Qi and concerns extending the known LCP characterization.
- Research question: Two counterexamples show that the proposed P-tensor characterization does not hold for tensor complementarity problems.The paper’s answer to the question is negative.
- Main contributions: When A is a P tensor, the TCP solution set is nonempty and compact, although solutions need not be unique.The paper distinguishes solvability and compactness from global uniqueness.
- Main contributions: The paper introduces strong P tensors and shows that their corresponding TCPs have the GUS-property.Strong P tensors provide a related structured-tensor class with the desired uniqueness and solvability property.
- Main contributions: The strong P tensors form a proper subset of the P tensors, while many P-tensor results remain valid for strong P tensors.The paper is organized around preliminaries, counterexamples and compactness, strong P tensors, and conclusions.
2 Preliminaries
This section defines complementarity problems and the tensor and linear specializations, introduces tensor eigenvalue terminology and structured tensor classes, and states the GUS question studied later.
- Complementarity problems: A complementarity problem CP(F) seeks x ∈ R^n satisfying nonnegativity and complementarity conditions involving F(x).The supplied passages state the complementarity conditions as x ≥ 0, F(x) ≥ 0, and x^T F(x) = 0.
- Linear and tensor complementarity: LCP(q, A) is obtained when F(x) = Ax + q, whereas TCP(q, A) is obtained when F(x) = Ax^(m−1) + q.The tensor mapping uses an mth-order n-dimensional tensor A.
- Tensor eigenvalues: An H-eigenvalue and H-eigenvector satisfy a tensor eigenvalue relation for a nonzero vector, while a Z-eigenpair additionally uses x^T x = 1.The supplied passages introduce both eigenvalue notions and explicitly state the normalization for Z-eigenvectors.
- Structured tensors and functions: The section reviews structured tensors, including P tensors, strictly semi-positive tensors, and R-tensors, together with P-functions and uniform P-functions.It states that every uniform P-function is a P-function and that every P tensor is strictly semi-positive and an R-tensor.
- Global uniqueness and solvability: TCP(q, A) has the GUS-property when it has a unique solution for every q ∈ R^n.The analogous LCP property holds if and only if A is a P-matrix.
- Global uniqueness and solvability: The central preliminary question is whether TCP(q, A) has the GUS-property if and only if A is a P tensor.The question was proposed by Song and Qi and is answered in the next section.
3 Answer to Q1
The paper answers Q1 negatively: a P tensor does not guarantee the GUS-property for the associated TCP. It proves instead that P-tensor TCP solution sets are nonempty and compact, and gives a separate GUS example outside the P-tensor class.
- GUS beyond P tensors: A separate constructed TCP has a unique solution for every q ∈ R^2 and therefore has the GUS-property, while its tensor is not a P tensor.The paper uses the nonexistence of odd-order P tensors to establish the latter point for the example.
- Counterexamples: The constructed tensor is a P tensor because every nonzero x has an index i satisfying x_i(Ax^3)_i > 0.
- Solution-set properties: For every q ∈ R^n, TCP(q, A) with a P tensor A has a nonempty and compact solution set.Compactness is established by proving boundedness and closedness of the solution set.
- Solution-set properties: Boundedness follows by contradiction: an unbounded solution sequence would produce a nonzero solution of TCP(0, A), contradicting strict semi-positivity of P tensors.
- Solution-set properties: Closedness follows because limits of solution sequences preserve nonnegativity, the tensor inequality, and the complementarity condition.
4 Strong P Tensor and Related Properties
The paper introduces strong P tensors, defined through P-functions, and proves that their tensor complementarity problems have the GUS-property. Strong P tensors form a proper subclass of P tensors and inherit several structural properties.
- Definition and motivation: A strong P tensor is defined as a tensor for which F(x) = Ax^(m−1) + q is a P-function.This definition enables results and methods associated with P-functions to be applied to strong P tensors.
- GUS-property: If A is a strong P tensor, then TCP(q, A) has a unique solution for every q ∈ R^n.The proof combines solvability for P tensors with the fact that P-functions yield at most one complementarity solution.
- Relationship with P tensors: Every strong P tensor is a P tensor, but the converse does not hold.The paper gives an example of a P tensor that is not a strong P tensor, so strong P tensors are a proper subset of P tensors.
- Related properties: Strong P tensors are strictly semi-positive, R tensors, have positive H-eigenvalues and Z-eigenvalues, and have positive diagonal entries.These properties follow from the corresponding properties of P tensors.
- Related properties: Every principal sub-tensor of a strong P tensor is also a strong P tensor.The result is established by embedding vectors from the principal sub-tensor into the full-dimensional space.
5 Conclusions
The paper gives two counterexamples that reject the proposed equivalence between P tensors and the GUS-property, while establishing compact nonempty solution sets for P tensors and GUS solvability for strong P tensors. It concludes that further strong-P-tensor properties may be studied using P-function results.
- Main conclusions: Two counterexamples show that P tensors neither imply the GUS-property nor are implied by it.This provides a negative answer to Question 6.3 of Song and Qi.
- Main conclusions: When A is a P tensor, the solution set of TCP(q, A) is nonempty and compact.
- Main conclusions: When A is a strong P tensor, TCP(q, A) has the GUS-property, and every strong P tensor is a P tensor.
- Future work: The paper suggests that additional strong P tensor properties can be studied using known P-function methods and results.