Liqun Qi

The Second-Order Augmented Zarankiewicz Number

Liqun Qi, Chunfeng Cui

The second-order Zarankiewicz number $z_{2} (m, n)$ and the biquadratic sum-of-squares rank $BSR (m, n)$ are related by the unconditional hierarchy \[ \mathrm{BSR}(m,n)\ \ge\ z_…

Exact Second-Order Zarankiewicz Numbers for Complete-Graph Incidence Families

Yannan Chen, Liqun Qi, Johan Löfberg

For $n \geq 6$ , $m = (2 n)$ , let the complete-graph incidence family on $K_{n}$ have the vertices of $K_{n}$ as columns, its edges as rows, and the incidence graph as one-edge graph. Th…

Is the Signed Zarankiewicz Number the Same as the Recursive-line Zarankiewicz Number?

Zhiwei Chen, Yannan Chen, Liqun Qi

Löfberg and Qi introduced the second order Zarankiewicz number $z_2$, the recursive-line Zarankiewicz number $z_{RL}$, and the signed Zarankiewicz number $z_{SL}$ for doubly…

Second Order Zarankiewicz Number

Johan Löfberg, Liqun Qi

We introduce the \emph{second order Zarankiewicz number} $z_{2} (m, n)$ for irreducible doubly simple biquadratic forms with $∣ E_{1} ∣ = z (m, n)$ , together with the intermediate recursive-li…

Exact and Asymptotic Values for Weak Limited Augmented Zarankiewicz Numbers in the $m \times 3$ Case

Liqun Qi, Johan Löfberg, Yannan Chen

We determine the exact weak limited augmented Zarankiewicz numbers $z_{w L} (m, 3)$ for all $m \geq 3$ : \[ z_{wL}(m,3)= \begin{cases} m+3+\left\lceil \dfrac{m}{2}\right\rceil+1, & 9\le…

A Weak Condition for Limited Augmented Zarankiewicz Numbers

Liqun Qi, Chunfeng Cui, Yi Xu

This paper introduces the weak augmented Zarankiewicz number $z_{w A} (m, n)$ and the weak limited augmented Zarankiewicz number $z_{w L} (m, n)$ , which are combinatorial extensions of t…

Three-Edges and the SOS Rank of Biquadratic Forms

Liqun Qi, Chunfeng Cui, Yi Xu

We extend the augmented bipartite graph framework for biquadratic sum-of-squares (SOS) ranks by introducing $3$ -edges -- triples of cells representing squares of three-term bilinea…

Optimization Models and Computational Bounds for Limited Augmented Zarankiewicz Numbers in the Incidence-Graph Family of Complete Graphs

Yi Xu, Gaohang Yu, Liqun Qi

Let $G_{1}$ denote the incidence graph of the complete graph $K_{q + 1}$ . We study limited augmented Zarankiewicz numbers in this family through structured 0--1 optimization models for…

Biquadratic SOS Rank and Augmented Zarankiewicz Number

Liqun Qi, Chunfeng Cui, Yi Xu

This paper introduces the concepts of the augmented Zarankiewicz number $z_{A} (m, n)$ and the limited augmented Zarankiewicz number $z_{L} (m, n)$ , which are natural combinatorial extensi…

A General Lower Bound for the Limited Augmented Zarankiewicz Number based upon Complete Graphs

Liqun Qi, Chunfeng Cui, Yi Xu

The limited augmented Zarankiewicz number $z_{L} (m, n)$ satisfies $BSR (m, n) \geq z_{L} (m, n) \geq z (m, n)$ , where $BSR (m, n)$ is the maximum SOS rank of $m\times…

Narrowing the Gap: SOS Ranks of $4 \times 3$ Biquadratic Forms and a Lower Bound of $8$

Yi Xu, Chunfeng Cui, Liqun Qi

We investigate the maximum sum-of-squares (SOS) rank of biquadratic forms in the critical case of $4 \times 3$ variables, where the general bounds are currently $7 \leq \mathrm{BSR…

Sum of Squares Rank of Biquadratic Forms and The Zarankiewicz Number

Chunfeng Cui, Liqun Qi, Yi Xu

Denote the maximum sos rank of $m \times n$ sum of squares (SOS) biquadratic forms by $B S R (m, n)$ . In this paper, we show that $B S R (m, n) \geq z (m, n)$ and conjecture that $BSR(m, n…

On the SOS Rank of Simple and Diagonal Biquadratic Forms

Yi Xu, Chufeng Cui, Liqun Qi

We study the sum-of-squares (SOS) rank of simple and diagonal biquadratic forms. For simple biquadratic forms in $3 \times 3$ variables, we show that the maximum SOS rank is exactl…

Parallelizable Riemannian Alternating Direction Method of Multipliers for Non-convex Pose Graph Optimization

Xin Chen, Chunfeng Cui, Deren Han +1

Pose graph optimization (PGO) is fundamental to robot perception and navigation systems, serving as the mathematical backbone for solving simultaneous localization and mapping (SLA…

Sum of Squares Decompositions and Rank Bounds for Biquadratic Forms

Liqun Qi, Chunfeng Cui, Yi Xu

We study SOS properties of biquadratic forms. For the class of partially symmetric biquadratic forms, we establish necessary and sufficient conditions for positive semi-definitenes…

Sum of Squares Decompositions for Structured Biquadratic Forms

Yi Xu, Chunfeng Cui, Liqun Qi

This paper studies sum-of-squares (SOS) representations for structured biquadratic forms. We prove that diagonally dominated symmetric biquadratic tensors are always SOS. For the s…

math.NA2025

The Power Method for Non-Hermitian Dual Quaternion Matrices

Hao Yang, Liqun Qi, Chunfeng Cui

This paper proposes a power method for computing the dominant eigenvalues of a non-Hermitian dual quaternion matrix (DQM). Although the algorithmic framework parallels the Hermitia…

Structured Symmetric Tensors

Liqun Qi, Chunfeng Cui, Yi Xu

In this paper, we study structured symmetric tensors. We introduce several new classes of structured symmetric tensors: completely decomposable (CD) tensors, strictly sum of square…

math.RA2025

Completely Positive Biquadratic Tensors

Liqun Qi, Chunfeng Cui, Haibin Chen +1

In this paper, we systemically introduce completely positive biquadratic (CPB) tensors and copositive biquadratic tensors. We show that all weakly CPB tensors are sum of squares te…

A Dual Quaternion Control Law for Formation Control of Multiple 3-D Rigid Bodies

Chunfeng Cui, Liqun Qi, Hao Chen +1

This paper studies the integrated position and attitude control problem for multi-agent systems of 3D rigid bodies. While the state-of-the-art method in [Olfati-Saber and Murray, 2…