AI 中文总结
研究四次或更高次多元多项式凸性判定难题，引入 SPBQ 形式多项式，分析其凸性与 SOS 凸性联系，构造特殊例子，研究相关优化问题，相比一般 SOS 方法有计算优势，可用于凸多项式回归和流体动力学。
AI 中文摘要
确定四次或更高次多元多项式是否非负和凸是强 NP 难问题。为缓解计算困难，提出平方和（SOS）凸性作为可处理的代数松弛，给出凸性可检验充分条件且可表示为半定规划（SDP）。本文引入四次多项式的结构化子类——可分离和双二次（SPBQ）形式，系统分析该类中凸性与 SOS 凸性的联系。具体表明当相关双二次形式为\(n\times2\)时，每个凸 SPBQ 多项式必为 SOS 凸。构造了一个\(3\times3\)双二次形式的显式 SPBQ 例子，它是凸的但不是 SOS 凸。最后研究了涉及 SPBQ 多项式的无约束和约束优化问题，与一般基于 SOS 的方法相比有显著计算优势，并说明了其在凸多项式回归和流体动力学中的应用。
英文摘要
Determining whether multivariate polynomials of degree four or higher are nonnegative and convex is a strongly NP-hard problem. To mitigate these computational difficulties, sum-of- squares (SOS) convexity has been proposed as a tractable algebraic relaxation that yields a checkable sufficient condition for convexity and can be expressed as a semidefinite program (SDP). In this work, we introduce a structured subclass of quartic polynomials, called the Sum of Separable and Biquadratic (SPBQ) forms, and conduct a systematic analysis of the connection between convexity and SOS-convexity within this class. Specifically, we show that every convex SPBQ polynomial is necessarily SOS-convex when the associated biquadratic form has size n x 2. We then construct an explicit SPBQ example with a 3 x 3 biquadratic form that is convex but fails to be SOS-convex. Finally, we examine both unconstrained and constrained optimization problems involving SPBQ polynomials, demonstrate notable computational benefits compared to general SOS-based methods, and illustrate their use in convex polynomial regression and fluid dynamics.
