arXiv subjects

Liqun Qi

Publications and source records attributed to Liqun Qi.

At least 19 recordsLinked to original sources

arXiv2026

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

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 simple biquadratic forms. It was shown that \[ z_2(m,n)\ge z_{SL}(m,n)\ge z_{RL}(m,n) \] for all $m$ and $n$. However, there was no evidence that there exist particular $m$ and $n$ such that $z_{SL}(m,n)>z_{RL}(m,n)$. The motivation for introducing $z_{SL}$ was as follows: during the study of the exceptional case $m=15$, $n=6$, Löfberg and Qi showed that \[ z_2(15,6)=z_{SL}(15,6)=60, \] but the exact value of $z_{RL}(15,6)$ was unknown then. In this paper we show that \[ z_{RL}(15,6)=60. \] This eliminates the motivation for introducing $z_{SL}$. Whether $z_{SL}(m,n)=z_{RL}(m,n)$ in general remains an open problem. Recently, Lebedev presented an explicit construction separating the augmented Zarankiewicz number $z_A$ from the limited augmented Zarankiewicz number $z_L$ at $m=n=1893$. We hope that the separation problem for $z_{SL}$ and $z_{RL}$ can also be solved. We also present the exact values of $z_{RL}(m,6)$ for $6\le m\le 16$.

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arXiv2026

Second Order Zarankiewicz Number

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-line and signed parameters $z_{RL}(m,n)$ and $z_{SL}(m,n)$. They satisfy the unconditional hierarchy \[ \operatorname{BSR}(m,n) \ge z_2(m,n) \ge z_{SL}(m,n) \ge z_{RL}(m,n) \ge z_{wL}(m,n) \ge z(m,n), \] where $z_{RL}$ is defined by the strengthened recursive rectangle criterion $(RW3^+)$ together with the conditions $(S)$ and $C_4$-freeness of $G_1$, and $z_{SL}$ is defined by the signed criterion $(RW3^\pm)$. We show that $(RW3^+)$ is sound and strictly weaker than the literal weak cross-cell test $(W3)$ on the weak-admissible class, and that $(RW3^\pm)$ is likewise sound. At the smallest four-column cases we obtain \[ z_2(5,4)=z_{SL}(5,4)=z_{RL}(5,4)=13>12=z_{wL}(5,4), \] and \[ z_2(6,4)=z_{SL}(6,4)=z_{RL}(6,4)=16>14=z_{wL}(6,4). \] In three columns this yields \[ z_2(m,3)=z_{RL}(m,3)=2m \qquad \text{for all } m\ge 3, \] with strict separation from $z_{wL}(m,3)$ for every $m\ge10$, and exact gap $\lfloor(m-3)/3\rfloor$ for $m\ge16$. Further finite computations give $z_{RL}(5,5)=17$, $z_{RL}(7,4)=19$, and $z_{RL}(7,7)\ge32>28=z_{wL}(7,7)$. Along $N=2p$ with $p$ an odd prime, we obtain the cubic asymptotic separation \[ z_2\!\left(\binom{N}{2},N\right)-z_{wL}\!\left(\binom{N}{2},N\right)\ge \left(\frac{1}{16}-o(1)\right)N^3. \] For the exceptional incidence case $p=3$ we prove $z_{SL}(15,6)=z_2(15,6)=60$. Concurrent work of Chen and Chen, using the recursive-line definition introduced here, incorporates the finite recursive-line values of this manuscript and establishes further exact four-column values together with an eventual formula for all $m\ge15$. The conjectural equality $z_2=z_{RL}$ is supported by the exact two-column, three-column, four-column, and odd-prime incidence families.

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arXiv2026

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

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 admissible 2-edge augmentation. For the original framework, we combine exact ILP formulations for the smallest instances with constructive search followed by exact admissibility verification for larger instances. This yields \[ {z_L(6,4)=14,\qquad z_L(10,5)=26,\qquad z_L(15,6)\ge 43,\qquad z_L(21,7)\ge 64,\qquad z_L(28,8)\ge 88.} \] The first two values are exact, whereas the latter three are rigorous lower bounds obtained from explicitly verified admissible families. We then formulate the global weak problem determined by $(S),(W2),(W2'),(W3)$. On the larger incidence-family cases, the corresponding global exact optimization problem is currently computationally out of reach, and our computations provide the certified lower bounds \[ {z_{WL}(6,4)=14,\quad z_{WL}(10,5)\ge 29,\quad z_{WL}(15,6)\ge 46,\quad z_{WL}(21,7)\ge 65,\quad z_{WL}(28,8)\ge 90.} \] On the fourteen literal-grid benchmarks from $(4,3)$ through $(7,7)$, by contrast, the global weak problem is solved exactly. Across this exact benchmark block, the weak framework is never worse than the strong framework and is strictly better in several cases. These computations improve the corresponding classical Zarankiewicz numbers and therefore strengthen available lower bounds for $\operatorname{BSR}(m,n)$ within this family.

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arXiv2026

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

We determine the exact weak limited augmented Zarankiewicz numbers $z_{wL}(m,3)$ for all $m\ge 3$: \[ z_{wL}(m,3)= \begin{cases} m+3+\left\lceil \dfrac{m}{2}\right\rceil+1, & 9\le m\le 15,\\[2mm] m+3+\left\lfloor \dfrac{2m-4}{3}\right\rfloor, & m\ge 16, \end{cases} \] with $z_{wL}(3,3)=6$, $z_{wL}(4,3)=8$, and $z_{wL}(m,3)=2m$ for $5\le m\le 9$. In particular, \[ \lim_{m\to\infty} \frac{z_{wL}(m,3)}{m} = \frac{5}{3}. \] The proof is fully analytic, relying on a uniform base classification, two constructive lower-bound families (staircase and $5m/3$), and a sharp upper-bound argument based on a peeling lemma and the analysis of two W2-sensitive boundary cases. Numerical MILP computations were used only as proof-mining tools to identify the structural lemmas; the final theorem is unconditional. We also extend the known range of the original limited numbers $z_L(m,3)$ through $m=13$, where the gap to $z_{wL}(m,3)$ is only 2 or 3.

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arXiv2026

A Weak Condition for Limited Augmented Zarankiewicz Numbers

This paper introduces the weak augmented Zarankiewicz number $z_{wA}(m,n)$ and the weak limited augmented Zarankiewicz number $z_{wL}(m,n)$, which are combinatorial extensions of the classical Zarankiewicz number obtained by relaxing the original admissibility conditions for augmented bipartite graphs. We show that the resulting weak framework still guarantees irreducibility of the associated doubly simple biquadratic forms, with SOS rank equal to the total number of edges. This yields the inequality chain \[ \mathrm{BSR}(m,n) \geq z_{wA}(m,n) \geq z_{wL}(m,n) \geq z_L(m,n) \geq z(m,n). \] We provide three complementary constructions demonstrating the power of the weak framework. First, a $5\times 3$ construction using degenerate 2-edges yields $z_{wL}(5,3)\ge 10>9=z_L(5,3)$, giving $\mathrm{BSR}(5,3)\ge 10$. Second, a $15\times 6$ construction on the incidence graph of $K_6$ with 14 nondegenerate 2-edges gives $z_{wL}(15,6)\ge 44>43$, improving the previously known bound. Third, a critical $6\times 3$ construction with complementary 2-cycles gives $z_{wL}(6,3)\ge 12>11=z_L(6,3)$, yielding $\mathrm{BSR}(6,3)\ge 12$ and demonstrating that complementary 2-cycles are safe.

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arXiv2026

Three-Edges and the SOS Rank of Biquadratic Forms

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 bilinear forms. We define suitable generalized cycle-free conditions that are purely combinatorial yet sufficient to guarantee that the SOS rank equals the total number of edges, carefully distinguishing occupation by $1$/$2$-edges from occupation by $3$-edges. The main theorem states that for any generalized cycle-free augmented bipartite graph $G$ satisfying the simplicity condition (S), the associated triply simple biquadratic form $P_G$ satisfies $\operatorname{sos}(P_G) = |E_1| + |E_2| + |E_3|$. The proof extends the orthogonality method with a novel trick: when a $2$-edge and a $3$-edge interact, the $3$-edge condition must be invoked rather than the $2$-edge condition. We give three applications. A $5 \times 3$ construction with two $2$-edges, inadmissible under the original definition, is admissible under our new definition, yielding $z_{3L}(5,3) \ge 10$ and improving $z_L(5,3)=9$. A $10 \times 5$ graph using a column-fully-degenerate $3$-edge gives $z_{3L}(10,5) \ge 27$, separating it from $z_L(10,5)=26$. A $15 \times 6$ graph using a half-row-degenerate $3$-edge improves the lower bound for $\operatorname{BSR}(15,6)$ from $43$ to $44$. These are the first explicit applications of $3$-edges to obtain improved lower bounds for $\operatorname{BSR}(m,n)$, and the $5 \times 3$ example demonstrates the power of the refined conditions.

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arXiv2026

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

The limited augmented Zarankiewicz number $z_L(m,n)$ satisfies $\operatorname{BSR}(m,n)\ge z_L(m,n)\ge z(m,n)$, where $\operatorname{BSR}(m,n)$ is the maximum SOS rank of $m\times n$ biquadratic forms and $z(m,n)$ is the classical Zarankiewicz number. Our main result is a general lower bound for $z_L(m,n)$ based on the incidence graph of the complete graph $K_{4t}$. For every integer $t\ge 1$, let $m = \binom{4t}{2}$ and $n = 4t$. Then $$ \operatorname{BSR}(m,n) \;\ge\; z_L(m,n) \;\ge\; 2\binom{4t}{2} + 4t^2 - 2t. $$ Since $z = 2\binom{4t}{2} = Θ(t^2)$, the gap satisfies $z_L - z \ge 4t^2 - 2t = Θ(t^2) = Θ(m)$, i.e., it grows linearly in $m$. Moreover, $$ \frac{z_L - z}{z} \;\ge\; \frac{4t^2}{16t^2 - 4t} \;\longrightarrow\; \frac{1}{4} \quad \text{as } t\to\infty, $$ so the gap is asymptotically at least $25\%$ of $z$ -- a non-negligible constant fraction. For $t=1$ we obtain $z_L(6,4)\ge 14$, and we prove that this bound is tight, i.e., $z_L(6,4)=14$. For $t=2$ and $t=3$ we obtain $z_L(28,8)\ge 68$ and $z_L(66,12)\ge 162$, respectively, improving previously known bounds. We also determine the exact values of $z_L(m,n)$ for $5\times3$ and $5\times4$: $z_L(5,3)=9$ and $z_L(5,4)=12$. These results serve as base cases for a \emph{lifting method} that constructs admissible limited augmented graphs on $(m+1)\times(n+1)$ from optimal ones on $m\times n$. Applying this method, we obtain new lower bounds: $z_L(6,3)\ge 10$ and $z_L(6,5)\ge 17$. For $5\times 5$ we establish a new lower bound $z_L(5,5)\ge 15$, improving the previously known bound. By a direct construction, we have $z_L(6,6)\ge 20$.

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arXiv2026

Biquadratic SOS Rank and Augmented Zarankiewicz Number

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 extensions of the classical Zarankiewicz number. These numbers arise from augmented bipartite graphs that may contain both standard edges (1-edges) and pairs of edges representing squares of binomials (2-edges). The main theoretical result establishes the inequality chain $\operatorname{BSR}(m, n) \geq z_A(m, n) \geq z_L(m, n) \geq z(m, n)$, linking the maximum biquadratic sum-of-squares (SOS) rank to these extremal graph parameters. We determine the exact values of $z_L(m, n)$ for the cases $(m,2)$, $(3,3)$, $(4, 3)$ and $(4,4)$, and provide new lower bounds for the cases $(5,3)$, $(5,4)$, and $(5,5)$. These results yield improved lower bounds for the maximum SOS rank of biquadratic forms, demonstrating that $z_L(m,n)$ can exceed the classical Zarankiewicz number, thereby offering a refined combinatorial perspective on the SOS rank problem.

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arXiv2026

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

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 $7 \leq \mathrm{BSR}(4,3) \leq 11$. By analyzing two important structured subclasses, we obtain exact determinations and improved upper bounds that significantly narrow this gap. For simple biquadratic forms those containing only distinct terms of the type $x_i^2 y_j^2$ we prove that the maximum achievable SOS rank is exactly 7, a value attained by a form corresponding to a $C_4$-free bipartite graph with the maximum number of edges. This settles the question for simple forms. For $y$-deficient biquadratic forms a class introduced here that permits cross terms among two of the three $y$-variables while the third appears only in pure square terms we prove an upper bound of $9$ by combining Calderön's theorem on $m\times 2$ forms with the known value $\mathrm{BSR}(4,2) = 5$. Our main result is a constructive proof that $\mathrm{BSR}(4,3) \geq 8$. We present an explicit non-simple, non-deficient $4\times 3$ biquadratic form and prove it requires exactly eight squares, thereby improving the general lower bound. This shows that any form achieving a rank higher than $8$ must possess a more complex algebraic structure, and it reduces the search space for determining the true value of $\mathrm{BSR}(4,3)$. Connections to Zarankiewicz numbers, extremal graph theory, and classical results on sums of squares are highlighted throughout.

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arXiv2026

Sum of Squares Rank of Biquadratic Forms and The Zarankiewicz Number

Denote the maximum sos rank of $m \times n$ sum of squares (SOS) biquadratic forms by $BSR(m, n)$. In this paper, we show that $BSR(m, n) \ge z(m, n)$ and conjecture that $BSR(m, n) = z(m, n)$, where $z(m, n)$ is the Zarankiewicz number. Our result coincides with the existing results for $m = 2$, $n = 2$, and $m = n = 3$, and is superior to other previously known lower bounds. Our result also connects graph theory and SOS polynomial theory.

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arXiv2026

On the SOS Rank of Simple and Diagonal Biquadratic Forms

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 exactly $6$, attained by a specific six-term form. We further prove that for any $m \ge 3$, there exists an $m \times m$ simple biquadratic form whose SOS rank is exactly $2m$. Moreover, we show that for all $m, n \ge 3$, the maximum SOS rank over $m \times n$ simple biquadratic forms is at least $m+n$, which implies $\mathrm{BSR}(m,n) \ge m+n$. For diagonal biquadratic forms with nonnegative coefficients, we prove an SOS rank upper bound of $7$, improving the general bound of $8$ for $3 \times 3$ forms. These results provide new lower and upper bounds on the worst-case SOS rank of biquadratic forms and highlight the role of structure in reducing the required number of squares.

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arXiv2026

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

Pose graph optimization (PGO) is fundamental to robot perception and navigation systems, serving as the mathematical backbone for solving simultaneous localization and mapping (SLAM). Existing solvers suffer from polynomial growth in computational complexity with graph size, hindering real-time deployment in large-scale scenarios. In this paper, by duplicating variables and introducing equality constraints, we reformulate the problem and propose a Parallelizable Riemannian Alternating Direction Method of Multipliers (PRADMM) to solve it efficiently. Compared with the state-of-the-art methods that usually exhibit polynomial time complexity growth with graph size, PRADMM enables efficient parallel computation across vertices regardless of graph size. Crucially, all subproblems admit closed-form solutions, ensuring PRADMM maintains exceptionally stable performance. Furthermore, by carefully exploiting the structures of the coefficient matrices in the constraints, we establish the global convergence of PRADMM under mild conditions, enabling larger relaxation step sizes within the interval $(0,2)$. Extensive empirical validation on two synthetic datasets and multiple real-world 3D SLAM benchmarks confirms the superior computational performance of PRADMM.

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arXiv2026

Biquadratic Cauchy Tensors and Spherical Biquadratic Polynomial Programming

This paper addresses biquadratic polynomial programming (BPP), an NP-hard optimization problem closely related to biquadratic tensors. We first establish several necessary and sufficient conditions for the positive semi-definiteness and positive definiteness of biquadratic Cauchy tensors. Leveraging the structured properties of these tensors, we then prove that the BPP and its equivalent multilinear formulation share the same set of optimal solutions. This result allows us to establish the global sequence convergence of the proximal alternating minimization (PAM) algorithm via the Kurdyka- Lojasiewicz (KL) property, extending the analysis in [8]. Furthermore, by reformulating the equivalent multilinear problem as an unconstrained optimization model, we enable the analysis of its KL exponent and derive an explicit expression for the convergence rate of PAM. Finally, numerical experiments are conducted on both biquadratic Cauchy tensors and general biquadratic tensor instances to evaluate the efficiency, stability, and practical performance of the proposed algorithm.

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arXiv2026

Sum of Squares Decompositions and Rank Bounds for Biquadratic Forms

We study SOS properties of biquadratic forms. For the class of partially symmetric biquadratic forms, we establish necessary and sufficient conditions for positive semi-definiteness and prove that every PSD partially symmetric biquadratic form is a sum of squares of bilinear forms. This extends the known result for fully symmetric biquadratic forms. We describe an efficient computational procedure for constructing SOS decompositions, exploiting the Kronecker-product structure of the associated matrix representation. We introduce simple biquadratic forms. For $m \ge 2$, we present a $m \times 2$ PSD biquadratic form and show that it can be expressed as the sum of $m+1$ squares, but cannot be expressed as the sum of $m$ squares. This provides a lower bound for sos rank of $m \times 2$ biquadratic forms, and shows that previously proved results that a $2 \times 2$ PSD biquadratic form can be expressed as the sum of three squares, and a $3 \times 2$ PSD biquadratic form can be expressed as the sum of four squares, are tight. We also present an $3 \times 3$ SOS biquadratic form, which can be expressed as the sum of six squares, but not the sum of five squares.We present a $2 \times 2$ PSD biquadratic form, and show that it can be expressed as the sum of three squares, but cannot be expressed as the sum of two squares. Furthermore, we present a $3 \times 2$ PSD biquadratic form, and show that it can be expressed as the sum of four squares, but cannot be expressed as the sum of three squares. These show that previously proved results that a $2 \times 2$ PSD biquadratic form can be expressed as the sum of three squares, and a $3 \times 2$ PSD biquadratic form can be expressed as the sum of four squares, are tight. Moreover, we establish a universal upper bound SOS-rank$(P) \le mn-1$ for any SOS biquadratic form, which improves the trivial bound $mn$ and is tight in small dimensions.

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arXiv2025

Sum of Squares Decompositions for Structured Biquadratic Forms

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 special case of symmetric biquadratic forms, we establish necessary and sufficient conditions for positive semi-definiteness of monic symmetric biquadratic forms, characterize the geometry of the corresponding PSD cone as a convex polyhedron, and prove that every such PSD form is SOS for any dimensions $m$ and $n$. We also formulate conjectures regarding SOS representations for symmetric M-biquadratic tensors and symmetric $\mathrm{B}_{0}$-biquadratic tensors, discussing their likelihood and potential proof strategies. Our results advance the understanding of when positive semi-definiteness implies sum-of-squares decompositions for structured biquadratic forms.

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arXiv2025

The Power Method for Non-Hermitian Dual Quaternion Matrices

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 Hermitian case, the theoretical analysis is substantially more complex since a non-Hermitian dual matrix may possess no eigenvalues or infinitely many eigenvalues. Besides, its eigenvalues are not necessarily dual numbers, leading to non-commutative behavior that further complicates the analysis. We first present a sufficient condition that ensures the existence of an eigenvalue whose standard part corresponds to the largest magnitude eigenvalue of the standard part matrix. Under a stronger condition, we then establish that the sequence generated by the power method converges linearly to the strict dominant eigenvalue and its associated eigenvectors. We also verify that this condition is necessary. The key to our analysis is a new Jordan-like decomposition, which addresses a gap arising from the lack of a conventional Jordan decomposition for non-Hermitian dual matrices. Our framework readily extends to non-Hermitian dual complex and dual number matrices. We also develop an adjoint method that reformulates the eigenvalue problem into an equivalent form for dual complex matrices. Numerical experiments on non-Hermitian DQMs are presented to demonstrate the efficiency of the power method.

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arXiv2025

The SOS Rank of Biquadratic Forms

In 1973, Calderón proved that an $m \times 2$ positive semidefinite (psd) biquadratic form can always be expressed as the sum of ${3m(m+1) \over 2}$ squares of quadratic forms. Very recently, by applying Hilbert's theorem on ternary quartics, we proved that a $2 \times 2$ psd biquadratic form can always be expressed as the sum of three squares of bilinear forms. This improved Calderón's result for $m=2$, and left the sos (sum-of-squares) rank problem of $m \times 2$ biquadratic forms for $m \ge 3$ to further exploration. In this paper, we show that an $3 \times 2$ psd biquadratic form can always be expressed as four squares of bilinear forms. We make a conjecture that an $m \times 2$ psd biquadratic form can always be expressed as $m+1$ squares of bilinear forms.

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arXiv2025

Structured Symmetric Tensors

In this paper, we study structured symmetric tensors. We introduce several new classes of structured symmetric tensors: completely decomposable (CD) tensors, strictly sum of squares (SSOS) tensors and SOS$^*$ tensors. CD tensors have applications in data analysis and signal processing. Complete Hankel tensors are CD tensors. SSOS tensors are defined as SOS tensors with a positive definite Gram matrix, ensuring structural stability under perturbations. The SOS$^*$ cone is defined as the dual cone of the SOS tensor cone, with characterizations via moment matrices and polynomial nonnegativity. We study the relations among completely positive (CP) cones, CD cones, sum of squares (SOS) cones, positive semidefinite (PSD) cones and copositive (COP) cones. We identify the interiors of PSD, SOS, CP, COP and CD cones for even-order tensors. These characterizations are crucial for interior-point methods and stability analysis in polynomial and tensor optimization. We generalize the classical Schur product theorem to CD and CP tensors, including the case of strongly completely decomposable (SCD) and strongly completely positive (SCP) tensors. We identify equivalence between strictly CD (SCD) and positive definite (PD) for CD tensors. Furthermore, we give an example of a PSD but not SOS Hankel tensor. This answers an open question raised in the literature.

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