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Biquadratic Cauchy Tensors and Spherical Biquadratic Polynomial Programming

Haibin Chen, Yixuan Chen, Liqun Qi

TL;DR

This work studies biquadratic polynomial programming (BPP), an NP-hard nonconvex optimization problem arising from unit-sphere constraints. It leverages the structure of biquadratic tensors, especially biquadratic Cauchy tensors, to derive necessary and sufficient conditions for positive semidefiniteness and positive definiteness, and to establish that the BPP is equivalent to a multilinear reformulation with the same optima. The authors introduce a proximal alternating minimization (PAM) algorithm, prove global convergence via the Kurdyka-Łojasiewicz property, and derive the KL exponent to obtain an explicit convergence rate by recasting the multilinear problem as an unconstrained model. Extensive numerical experiments on both structured (biquadratic Cauchy) and general biquadratic tensors demonstrate PAM’s efficiency and robustness, compare it to MBI and ADMM, and illustrate parameter sensitivities and practical performance. The results provide a theoretically grounded, scalable approach to a broad class of high-order polynomial optimizations under spherical constraints with potential impact in physics, signal processing, and multiway data analysis.

Abstract

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.

Biquadratic Cauchy Tensors and Spherical Biquadratic Polynomial Programming

TL;DR

This work studies biquadratic polynomial programming (BPP), an NP-hard nonconvex optimization problem arising from unit-sphere constraints. It leverages the structure of biquadratic tensors, especially biquadratic Cauchy tensors, to derive necessary and sufficient conditions for positive semidefiniteness and positive definiteness, and to establish that the BPP is equivalent to a multilinear reformulation with the same optima. The authors introduce a proximal alternating minimization (PAM) algorithm, prove global convergence via the Kurdyka-Łojasiewicz property, and derive the KL exponent to obtain an explicit convergence rate by recasting the multilinear problem as an unconstrained model. Extensive numerical experiments on both structured (biquadratic Cauchy) and general biquadratic tensors demonstrate PAM’s efficiency and robustness, compare it to MBI and ADMM, and illustrate parameter sensitivities and practical performance. The results provide a theoretically grounded, scalable approach to a broad class of high-order polynomial optimizations under spherical constraints with potential impact in physics, signal processing, and multiway data analysis.

Abstract

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.
Paper Structure (3 sections, 1 theorem, 5 equations)

This paper contains 3 sections, 1 theorem, 5 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Positive semi-definiteness of biquadtratic Cauchy Tensors

Key Result

theorem 1

Assume that a biquadratic Cauchy tensor ${\@fontswitch\mathcal{A}}$ has generating vectors ${\bf c} = (c_1,\cdots,c_m)^\top\in\mathbb{R}^m$ and ${\bf d} = (d_1,\cdots,d_n)^\top\in\mathbb{R}^n$. Then ${\@fontswitch\mathcal{A}}$ is positive semi-definite if and only if $c_i+d_j > 0, i\in[m], j\in[n]$.

Theorems & Definitions (1)

  • theorem 1