[Paper Review] P-Tensors, P$_0$-Tensors, and Tensor Complementarity Problem

Weiyang Ding, Ziyan Luo+1 more|arXiv (Cornell University)|Jul 24, 2015

Tensor decomposition and applications26 references18 citations

TL;DR

This paper introduces a homogeneous definition of P-tensors and P₀-tensors for even and odd-order tensors, extending the classical P-matrix concept to higher-order tensors. It proves that the tensor complementarity problem (TCP) with a P-tensor has a nonempty and compact solution set, establishing a foundational result for TCP theory and unifying key tensor classes like positive definite, M-tensors, and strictly diagonally dominant tensors with positive diagonals.

ABSTRACT

The concepts of P- and P$_0$-matrices are generalized to P- and P$_0$-tensors of even and odd orders via homogeneous formulae. Analog to the matrix case, our P-tensor definition encompasses many important classes of tensors such as the positive definite tensors, the nonsingular M-tensors, the nonsingular H-tensors with positive diagonal entries, the strictly diagonally dominant tensors with positive diagonal entries, etc. As even-order symmetric PSD tensors are exactly even-order symmetric P$_0$-tensors, our definition of P$_0$-tensors, to some extent, can be regarded as an extension of PSD tensors for the odd-order case. Along with the basic properties of P- and P$_0$-tensors, the relationship among P$_0$-tensors and other extensions of PSD tensors are then discussed for comparison. Many structured tensors are also shown to be P- and P$_0$-tensors. As a theoretical application, the P-tensor complementarity problem is discussed and shown to possess a nonempty and compact solution set.

Motivation & Objective

• To generalize the P-matrix concept to higher-order tensors using a homogeneous formulation applicable to both even and odd-order tensors.
• To define P-tensors and P₀-tensors in a way that preserves key properties of P-matrices and extends positive semidefinite (PSD) tensors to odd-order cases.
• To establish that the tensor complementarity problem (TCP) with a P-tensor has a nonempty and compact solution set, ensuring theoretical solvability.
• To characterize relationships between P₀-tensors and other extensions of PSD tensors, such as copositive, completely positive, and doubly nonnegative tensors.
• To demonstrate that important structured tensors—like positive definite, M-tensors, H-tensors, and strictly diagonally dominant tensors with positive diagonals—are special cases of P-tensors.

Proposed method

• Proposes a homogeneous definition of P-tensors via the condition: for every nonzero x ∈ ℝⁿ, there exists an index i such that xi(𝒜x^{m−1})_i > 0.
• Defines P₀-tensors analogously with weak inequality: xi(𝒜x^{m−1})_i ≥ 0 for all i, with strict inequality for at least one i.
• Uses homogeneity of the map x ↦ 𝒜x^{m−1} to analyze solution sets of the tensor complementarity problem (TCP) under P-tensor conditions.
• Applies a fixed-point lemma (Lemma 6.1) to show that the solution set of TCP(𝒜, q) is nonempty and compact when 𝒜 is a P-tensor.
• Establishes that P-tensors are a subclass of S-tensors by showing existence of x > 0 such that 𝒜x^{m−1} > 0.
• Analyzes tensor cones and their relationships to other special tensor classes, including PSD, copositive, and completely positive tensors.

Experimental results

Research questions

• RQ1 Can the P-matrix concept be extended to higher-order tensors in a way that applies to both even and odd-order tensors?
• RQ2 Does the tensor complementarity problem (TCP) with a P-tensor always admit a nonempty and compact solution set?
• RQ3 How do P₀-tensors relate to other extensions of positive semidefinite (PSD) tensors, especially in the odd-order case?
• RQ4 Are well-known structured tensors—such as positive definite, M-tensors, and strictly diagonally dominant tensors with positive diagonals—special cases of P-tensors?
• RQ5 Can P-tensors be used to characterize local optimality in multivariate functions via higher-order derivatives?

Key findings

• The tensor complementarity problem (TCP) with a P-tensor has a nonempty and compact solution set, ensuring theoretical solvability for all q.
• P-tensors are a proper subclass of S-tensors, as every P-tensor admits a strictly positive solution vector x > 0 such that 𝒜x^{m−1} > 0.
• Even-order symmetric positive semidefinite (PSD) tensors are exactly even-order symmetric P₀-tensors, extending the PSD class to odd-order via P₀-tensors.
• Many important structured tensors—such as positive definite tensors, nonsingular M-tensors, nonsingular H-tensors with positive diagonal entries, and strictly diagonally dominant tensors with positive diagonals—are special cases of P-tensors.
• The third-order derivative tensor of a C³ function is a P-tensor if the gradient and Hessian vanish at a point, implying that f(x + αd) > f(x) > f(x − αd) for small α > 0 and d > 0.
• P₀-tensors generalize the concept of PSD tensors to odd-order tensors, providing a natural extension for odd-order symmetric tensors.