Seminars on Numerical Algebra, Optimization and Data Sciences：The SOS Problem of Biquadratic Forms

：祁力群（香港理工大学）
：2025-12-31 16:00
：海韵园行政楼C503

报告人：祁力群（香港理工大学）

时  间：2025年12月31日16:00

地  点：海韵园行政楼C503

内容摘要:

A fundamental question at the intersection of algebra and optimization is whether a multivariate polynomial that is nonnegative everywhere (positive semi-definite, or PSD) can be written as a sum of squares (SOS) of polynomials. This talk focuses on the SOS Problem of a specific and important class of polynomials: Biquadratic Forms.

In 1973, Calderon proved that an m x 2 psd biquadratic form can always be expressed as the sum of squares (sos) of 3m(m+1)/2 quadratic forms. In 1975, Choi gave a concrete example of a 3 x 3 psd biquadratic form which is not sos. This gave a general picture of the SOS problem of biquadratic forms. Recently, in a series of papers, we systematically renew our knowledge on this problem. What we have done are as follow:

1. By applying Hilbert’s theorem, we proved that a 2 x 2 psd biquadratic form can always be expressed as the sum of three squares of bilinear forms,

2. By combining real analysis and algebraic geometry, we proved that a 3 x 2 psd biquadratic form can always be expressed as the sum of four squares of bilinear forms. We further made a conjecture that a m x 2 psd biquadratic form can always be expressed as the sum of m + 1 squares of bilinear forms, These systematically improved Calderon’s result.

3. We introduced symmetric biquadratic forms. Choi’s example is not a symmetric biquadratic form. We showed that all psd symmetric biquadratic forms are sos. This opens a new research direction.

4. We identified a number of psd structured biquadratic tensors are sos.

This also open a new research direction.

个人简介：

祁力群，香港理工大学应用数学系荣休教授，俄罗斯彼得罗夫斯卡亚科学与艺术研究院外籍院士。1968年获清华大学计算数学学士学位，1984年获美国威斯康星大学麦迪逊分校计算机科学博士学位，曾任教于清华大学、澳大利亚新南威尔士大学、香港城市大学等高校。祁力群教授长期从事非光滑优化、张量分析及计算数学研究，建立半光滑牛顿方法的超线性收敛性理论并提出高阶张量特征值概念并构建张量谱理论，相关成果获2010年中国运筹学会科学技术一等奖。祁力群教授在Mathematics of Computation, Mathematical Programming, Mathematics of Operations Research, SIAM Journal on Numerical Analysis, SIAM Journal on Optimization等期刊上发表论文380余篇，担任十多个国际期刊主编或编委，其著作《张量分析：谱理论和特殊张量》于2017年由SIAM出版，另一本书《张量特征值及其应用》于2018年由Springer出版。

 

联系人：白正简