Interval Double B-Tensors

• Interval double B-tensors are convex families of real tensors determined by two endpoint tensors, ensuring all members satisfy strict double B-tensor criteria including diagonal dominance and row sum conditions.
• They generalize interval matrix theory to high-order multilinear algebra, providing precise verification for applications in polynomial optimization and complementarity problems under uncertainty.
• Special structured cases, such as interval Z-tensors and circulant tensors, allow for simplified criteria that enhance computational efficiency in practical analytical contexts.

An interval double B-tensor is a convex family of real tensors of fixed order and dimension, parameterized by two endpoint tensors, with the property that every member satisfies the double B-tensor conditions. This concept generalizes interval matrix theory to high-order multilinear algebra and provides a foundational tool for analyzing polynomial optimization and complementarity problems under uncertainty. The main theoretical contributions include precise necessary and sufficient conditions for membership, simplifications for structured classes, and strong connections to established tensor-theoretic properties such as interval B-tensors, interval Z-tensors, and interval P-tensors (Ye et al., 18 Jan 2026).

1. Definition and Basic Properties

Let $T_{m,n}$ denote the space of real $m$th-order $n$-dimensional tensors. An interval tensor $\mathbb{T} = [\underline{T},\overline{T}]$ is the set

$$\mathbb{T} := \{ T \in T_{m,n} : \underline{T} \le T \le \overline{T} \}$$

where inequalities are entrywise. The lower and upper endpoint tensors $\underline{T}, \overline{T}$ uniquely define the interval. The extreme-point tensors of $\mathbb{T}$ are exactly those for which each entry equals either the lower or upper endpoint, so $\mathbb{T}$ is the convex hull of its extreme points.

A single tensor $A \in T_{m,n}$ is a double B-tensor if, for each $i \in [n]$, the following conditions hold:

• (a) Diagonal dominance: $a_{i\ldots i} > \gamma_i^+(A)$, where $$\gamma_i^+(A) = \max\{0, \max_{(i_2\ldots i_m) \neq (i\ldots i)} a_{ii_2\ldots i_m}\}$$.
• (b) Row sum inequality: $$a_{i\ldots i} - \gamma_i^+(A) \ge \sum_{(i_2\ldots i_m) \neq (i\ldots i)} (\gamma_i^+(A) - a_{ii_2\ldots i_m})$$.
• (c) Two-row product inequality for all distinct $i, j$:

$$\big(a_{i\ldots i} - \gamma_i^+(A)\big)\big(a_{j\ldots j} - \gamma_j^+(A)\big) > \Big( \sum_{(i_2\ldots i_m)\neq(i\ldots i)} (\gamma_i^+-a_{ii_2\ldots i_m}) \Big) \Big( \sum_{(j_2\ldots j_m)\neq(j\ldots j)} (\gamma_j^+-a_{jj_2\ldots j_m}) \Big)$$

An interval tensor $\mathbb{T}$ is an interval double B-tensor if every $T\in\mathbb{T}$ is a double B-tensor, or equivalently, all extreme-point tensors are double B-tensors [(Ye et al., 18 Jan 2026), Definition 2.9].

2. Characterization: Necessary and Sufficient Criteria

Determining interval double B-tensor membership reduces to endpoint inequalities (Theorem 4.2). Let $\mathbb{T} = [\underline{T},\overline{T}]$. $\mathbb{T}$ is an interval double B-tensor if and only if, for all $i,j\in[n]$, $(i_2,\ldots,i_m)\neq(i,\ldots,i)$, and $(j_2,\ldots,j_m)\neq(j,\ldots,j)$:

• (a) $$\underline{t}_{i\ldots i} > \max\{0,\;\overline{t}_{i\,i_2\ldots i_m}:(i_2\ldots i_m)\neq(i\ldots i)\}$$
• (b) (b1) $$\underline{t}_{i\ldots i} - \overline{t}_{i\,i_2\ldots i_m} \geq \Delta_i$$ where $$\Delta_i = \max\{0, \sum_{(k_2\ldots k_m)\ne(i\ldots i),(k_2\ldots k_m)\ne(i_2\ldots i_m)} [\overline{t}_{i\,i_2\ldots i_m} - \underline{t}_{i\,k_2\ldots k_m}]\}$$; (b2) $$\underline{t}_{i\ldots i}\geq \max\{0, -\sum_{(k_2\ldots k_m)\neq (i\ldots i)} \underline{t}_{i k_2\ldots k_m}\}$$
• (c) (c1) $$(\underline{t}_{i\ldots i}-\overline{t}_{i\,i_2\ldots i_m})(\underline{t}_{j\ldots j}-\overline{t}_{j\,j_2\ldots j_m}) > \Delta_i \Sigma_j$$; (c2) $$(\underline{t}_{i\ldots i}-\overline{t}_{i\,i_2\ldots i_m}) \underline{t}_{j\ldots j} > \Delta_i P_j$$; (c3) $$\underline{t}_{i\ldots i} \underline{t}_{j\ldots j} > P_i P_j$$, with
  • $$\Sigma_j = \max\{0,\sum_{(k_2\ldots k_m)\neq(j\ldots j),(k_2\ldots k_m)\neq(j_2\ldots j_m)} (\overline{t}_{j\,j_2\ldots j_m}-\underline{t}_{j\,k_2\ldots k_m})\}$$
  • $$P_j = \max\{0, -\sum_{(k_2\ldots k_m)\neq(j\ldots j)}\underline{t}_{j k_2\ldots k_m}\}$$

These conditions are both necessary and sufficient [(Ye et al., 18 Jan 2026), Theorem 4.2].

3. Structured Classes: Specialized Criteria

Significant simplifications arise for classes with additional structure:

• Interval Z-tensors: If all off-diagonal entries in any $T\in\mathbb{T}$ are nonnegative (interval Z-tensor), then $\mathbb{T}$ is an interval double B-tensor if and only if $\underline{T}$ is a double B-tensor itself (Proposition 4.16).
• Circulant interval tensors: If both endpoints are circulant, then interval double B, interval B, and interval B-tensor coincide. Verification reduces to checking only the first row (Proposition 4.20):
  • $$\sum_{i_2\ldots i_m} \underline{t}_{1\,i_2\ldots i_m} > 0$$
  • $$\underline{t}_{1\ldots 1} - \overline{t}_{1\,j_2\ldots j_m} > \sum_{(i_2\ldots i_m)\neq(1\ldots 1),(i_2\ldots i_m)\neq(j_2\ldots j_m)}(\overline{t}_{1\,i_2\ldots i_m} - \underline{t}_{1\,j_2\ldots j_m})$$ for each off-diagonal position $(j_2,\ldots,j_m)\neq(1,\ldots,1)$

This dramatically reduces complexity in highly symmetric or sparsely structured tensors (Ye et al., 18 Jan 2026).

4. Relations to Other Tensor Classes

Interval double B-tensors form a strict subclass of interval B-tensors (Proposition 4.12). The former property generally implies the latter, but the converse does not necessarily hold.

When the order $m$ is even and both endpoints are symmetric, every interval double B-tensor is an interval P-tensor (Corollary 4.8). That is, for every $T \in \mathbb{T}$ and every $x\neq 0$, there exists $i$ such that

$x_i^{m-1} (T x^{m-1})_i > 0.$

This result establishes that, under symmetry and even order, such interval families are guaranteed to have the strong positivity property essential in many optimization and variational inequality contexts.

5. Illustrative Example and Endpoint Reduction

A prototypical example is a $3\times 3\times 3$ interval tensor:

$$\underline{T}_{iii}=10,\qquad \underline{T}_{ijk}=0\ \textrm{for}\ i,j,k\textrm{ not all equal}$$

$$\overline{T}_{iii}=12,\qquad \overline{T}_{ijk}=1\ \textrm{for}\ i,j,k\textrm{ not all equal}$$

Checking the endpoint conditions suffices:

• (a) $10 > 1$
• (b) $10-1=9\geq 7$ (sum over other off-diagonals $7\times (1-0) = 7$), $10 \geq 0$
• (c) $(10-1)\times(10-1)=81>49=7\times 7$, etc.

All endpoint conditions are strictly satisfied, confirming the full family is an interval double B-tensor (Ye et al., 18 Jan 2026).

6. Applications and Analytical Implications

The extension of interval matrix theory to higher-order tensors via interval double B-tensors supplies explicit endpoint-driven verification criteria, enabling robust system analysis within polynomial optimization and complementarity frameworks under parametric uncertainty. In the presence of even order and symmetry, the guarantee of the interval P-tensor property facilitates positive definiteness and feasibility analysis for tensor complementarity problems (TCPs), and supports the tractable analysis of sums-of-squares relaxations in polynomial optimization.

In Z-tensor and circulant tensor settings, verifying interval double B-tensor conditions reduces computational effort dramatically, making the framework practical for structured problem instances (such as large-scale discretized PDEs or graph-based tensor data).

7. Connection to Eigenvalue Localization and Related Notions

Double B-tensors and double $\overline{B}$-tensors generate eigenvalue inclusion sets, refining classical Gerschgorin and Brauer-type estimates for tensors (Jin et al., 2015). The double $\overline{B}$-intervals, constructed analogously to classical matrix theory, yield bounds for all $H$-eigenvalues of even-order real symmetric tensors. These inclusion sets—constructed from diagonal, off-diagonal, and structured summations—are strictly contained within quasi-double $\overline{B}$ intervals, and thus supply increasingly precise eigenvalue localizations. The intersection of such intervals with Cassini-oval generalizations yields further refinement. This connection strengthens the spectral theoretical underpinning of double B- and interval double B-tensor theory, with critical implications for the positive definiteness and numerical range analysis of uncertain tensor systems (Jin et al., 2015).