Source-linked AI summary
Tensor Complementarity Problem and Semi-positive Tensors
Yisheng Song, Liqun Qi
TL;DR
The paper asks how tensor complementarity solution uniqueness relates to semi-positive tensors and how semi-positivity relates to copositivity for symmetric tensors. It proves the corresponding equivalences and concludes that strictly copositive symmetric tensors give a solution for every q ∈ R^n.
Problem
The paper investigates relationships between tensor complementarity solution uniqueness, semi-positive tensors, and copositive symmetric tensors.
Method
The paper proves equivalence theorems connecting semi-positivity with uniqueness or absence of nonzero tensor complementarity solutions, and connects symmetric semi-positivity with copositivity.
Results
A real tensor is semi-positive exactly when TCP(q, A) has a unique solution for q > 0, while strict semi-positivity corresponds to uniqueness for q ≥0; symmetric versions are equivalent to copositivity.
Takeaways & Limitations
For a strictly copositive symmetric tensor A, TCP(q, A) has a solution for every q ∈ R^n.
Abstract
from arXiv · showhide
The tensor complementarity problem $(\q, \mathcal{A})$ is to $$\mbox{ find } \x \in \mathbb{R}^n\mbox{ such that }\x \geq \0, \q + \mathcal{A}\x^{m-1} \geq \0, \mbox{ and }\x^\top (\q + \mathcal{A}\x^{m-1}) = 0.$$ We prove that a real tensor $\mathcal{A}$ is a (strictly) semi-positive tensor if and only if the tensor complementarity problem $(\q, \mathcal{A})$ has a unique solution for $\q>\0$ ($\q\geq\0$), and a symmetric real tensor is a (strictly) semi-positive tensor if and only if it is (strictly) copositive. That is, for a strictly copositive symmetric tensor $\mathcal{A}$, the tensor complementarity problem $(\q, \mathcal{A})$ has a solution for all $\q \in \mathbb{R}^n$.
1 Introduction
The paper extends complementarity-problem analysis from matrices to tensors, focusing on solution uniqueness and links between semi-positivity and copositivity. It establishes characterizations for tensor complementarity problems and symmetric tensors.
- 1 Introduction: Earlier matrix results connect semi-monotonicity with linear complementarity solutions and motivate analogous tensor results.The paper builds on established relationships for linear complementarity problems.
- 1 Introduction: The tensor complementarity problem is a structured nonlinear complementarity problem and a natural extension of the linear complementarity problem.It serves as the paper’s central problem framework.
- 1 Introduction: The paper studies relationships between unique tensor complementarity solutions and strictly semi-positive tensors.This is the main research focus stated in the introduction.
- 1 Introduction: Semi-positive tensors have nonnegative diagonal entries, while strictly semi-positive tensors have positive diagonal entries.These are preliminary structural properties developed in Section 2.
- 1 Introduction: A real tensor is semi-positive exactly when the tensor complementarity problem has no nonzero solution for q > 0, with the strict version using q ≥0.The introduction states these equivalent solution characterizations.
- 1 Introduction: For symmetric tensors, semi-positivity is equivalent to copositivity, and strict semi-positivity is equivalent to strict copositivity.Consequently, strictly copositive symmetric tensors yield solvable tensor complementarity problems for every q ∈ R^n.
2 Preliminaries
The preliminaries define real tensors, tensor-vector operations, semi-positivity, copositivity, Q-tensors, and principal subtensors. They also record structural properties used in the main results.
- 2 Preliminaries: The paper treats real mth-order n-dimensional tensors and defines tensor-vector multiplication through components of Ax^(m−1).The associated Ax^m is a homogeneous polynomial of degree m.
- 2 Preliminaries: A tensor is semi-positive or strictly semi-positive according to indexwise positivity conditions for every nonzero nonnegative vector.The definitions generalize semi-monotonicity concepts from matrices.
- 2 Preliminaries: A Q-tensor is defined by solvability of the tensor complementarity problem for every nonnegative q.Every strictly semi-positive tensor is a Q-tensor.
- 2 Preliminaries: Strictly semi-positive tensors have positive diagonal entries and specified positive entries associated with every index choice.These properties follow by testing the definition on particular nonnegative vectors.
- 2 Preliminaries: For symmetric tensors, copositivity and strict copositivity are characterized by nonnegative and positive values of Ax^m on nonnegative vectors, respectively.These polynomial characterizations support the later equivalence theorems.
- 2 Preliminaries: Every principal subtensor of a semi-positive tensor is semi-positive, and every principal subtensor of a strictly semi-positive tensor is strictly semi-positive.A principal subtensor is obtained by restricting all tensor indices to a selected subset.
3 Main results
The paper characterizes semi-positive tensors through uniqueness of tensor complementarity problem solutions and establishes equivalent copositivity conditions for symmetric tensors. It also derives existence results for complementarity problems with arbitrary vectors q in the strictly copositive symmetric case.
- A real tensor is semi-positive exactly when TCP(q, A) has a unique solution for every q > 0.
- The proof establishes the semi-positive characterization by ruling out nonzero TCP solutions and deriving semi-positivity from the absence of solutions to the associated system.
- A real tensor is strictly semi-positive exactly when TCP(q, A) has a unique solution for every q ≥ 0.
- For symmetric tensors, semi-positivity is equivalent to copositivity.
- For symmetric tensors, strict semi-positivity is equivalent to strict copositivity.
- A strictly copositive symmetric tensor yields a TCP solution for every q ∈ R^n.