Biquadratic Cauchy Tensors and Spherical Biquadratic Polynomial Programming

TL;DR

This paper studies biquadratic Cauchy tensors and their application to solving biquadratic polynomial programming, establishing conditions for positive definiteness, analyzing convergence of an optimization algorithm, and validating results through numerical experiments.

Contribution

It provides new conditions for positive definiteness of biquadratic Cauchy tensors and proves the convergence of the PAM algorithm for biquadratic polynomial programming.

Findings

Established necessary and sufficient conditions for positive definiteness.

Proved global convergence of the PAM algorithm.

Numerical experiments demonstrate efficiency and stability.

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.