14 papers
A Weak Condition for Limited Augmented Zarankiewicz Numbers
Liqun Qi, Chunfeng Cui, Yi Xu
This paper introduces the weak augmented Zarankiewicz number and the weak limited augmented Zarankiewicz number , 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 -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 denote the incidence graph of the complete graph . 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 and the limited augmented Zarankiewicz number , 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 satisfies , where is the maximum SOS rank of $m\times…
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…
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…
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…
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…
The SOS Rank of Biquadratic Forms
Liqun Qi, Chunfeng Cui, Yi Xu
In 1973, Calderón proved that an positive semidefinite (psd) biquadratic form can always be expressed as the sum of squares of quadratic forms. Ver…