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Exact and Asymptotic Values for Weak Limited Augmented Zarankiewicz Numbers in the Case
Liqun Qi, Johan Löfberg, Yannan Chen
We determine the exact weak limited augmented Zarankiewicz numbers for all : \[ z_{wL}(m,3)= \begin{cases} m+3+\left\lceil \dfrac{m}{2}\right\rceil+1, & 9\le…
Biquadratic SOS Rank and Augmented Zarankiewicz Number
Liqun Qi, Chunfeng Cui, Yi Xu
This paper introduces the concepts of the augmented Zarankiewicz number and the limited augmented Zarankiewicz number , which are natural combinatorial extensi…
Narrowing the Gap: SOS Ranks of Biquadratic Forms and a Lower Bound of
Yi Xu, Chunfeng Cui, Liqun Qi
We investigate the maximum sum-of-squares (SOS) rank of biquadratic forms in the critical case of 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 sum of squares (SOS) biquadratic forms by . In this paper, we show that 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 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…
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…
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…
Postive Semidefinite and Sum of Squares Biquadratic Polynomials
Chunfeng Cui, Liqun Qi, Yi Xu
Hilbert proved in 1888 that a positive semi-definite (PSD) homogeneous quartic polynomial of three variables always can be expressed as the sum of squares (SOS) of three quadratic…
Non-convex Pose Graph Optimization in SLAM via Proximal Linearized Riemannian ADMM
Xin Chen, Chunfeng Cui, Deren Han +1
Pose graph optimization (PGO) is a well-known technique for solving the pose-based simultaneous localization and mapping (SLAM) problem. In this paper, we represent the rotation an…