【报告摘要】Quaternions provide a natural representation for three-dimensional rotations and multichannel data, but their noncommutative multiplication makes polynomial optimization over quaternionic variables fundamentally different from its real and complex counterparts. In this talk, I will present a Moment–QSOS framework for global polynomial optimization over quaternions. A central difficulty is that quaternionic variables exhibit a hybrid algebraic structure: multiplication is noncommutative, while point evaluations satisfy additional polynomial identities inherited from the four commuting real coordinates of each quaternion. To capture this structure, we introduce quaternionic sums of squares together with scalar-identity ideals. We establish Archimedean Positivstellensätze for quaternionic polynomials, develop corresponding real-valued and quaternion-valued moment theories, and derive convergent Moment–QSOS hierarchies. Numerical experiments on quadratic, quartic, and sparse problems indicate that exploiting the quaternionic structure can substantially reduce computational cost while preserving relaxation quality. Applications to quaternion-based feature extraction and rotation synchronization will also be discussed.

【报告人简介】王杰，2012年本科毕业于中国科学技术大学，2017年博士毕业于中科院数学院，2017-2019年在北京大学从事博士后研究，2019-2021年在法国国家科学中心和Jean B. Lasserre教授合作从事大规模多项式优化的研究。入选中科院数学院“陈景润未来之星计划”、中国运筹学会青年人才发展专项。在SIOPT、SIAGA、MP、PRX、IEEE TAC、ACM TOMS等国际权威期刊发表论文40余篇，在World Scientific Press出版学术专著《Sparse Polynomial Optimization: Theory and Practice》。研究兴趣包括：多项式优化、半定规划、量子信息、量子多体计算等。