报 告 题 目：Normal Cones Intersection Rule and Optimality Analysis for Low-Rank Matrix Optimization with Affine Manifolds

主 讲 人：罗 自 炎

单 位：北京交通大学

时 间：10月19日10:30

腾 讯 ID: 773-288-722

摘 要：

The low-rank matrix optimization with affine manifolds (rank-MOA) aims to minimize a continuously differentiable function over a low-rank set intersecting with an affine manifold. In this talk, we will give the optimality analysis for rank-MOA. As a cornerstone, the intersection rule of the Fréchet normal cone to the feasible set of rank-MOA is established under some mild linear independence assumptions. Aided with the resulting explicit formulae of the underlying normal cones, the so-called F-stationary point and the F-stationary point of rank-MOA are investigated and the relationship with local/global minimizers are then revealed in terms of first-order optimality conditions. Furthermore, the second-order optimality analysis, including the necessary and the sufficient conditions, is proposed based on the second-order differentiation information of the model. All these results will enrich the theory of low-rank matrix optimization and give potential clues to designing efficient numerical algorithms for seeking low rank solutions. Meanwhile, two specific applications of rank-MOA are discussed to illustrate our proposed optimality analysis.

简 介：

罗自炎，女，北京交通大学数学与统计学院教授、博士生导师。曾为美国斯坦福大学、新加坡国立大学、英国南安普顿大学访问学者，香港理工大学研究助理。SCI论文40余篇 (ESI高被引论文2篇)，发表在《Math Program》《J Mach Learn Res》《IEEE Trans Signal Process》《SIAM J Matrix Anal Appl》等权威期刊。合作撰写SIAM出版社英文专著1部、中文著作1部；主持国家自然科学基金“面上”、“青年”基金项目、北京市自然科学基金重点项目等。2016年北京运筹学年会特邀大会报告; 2017年第十一届全国数学规划学术会议青年专题特邀报告; 2020年亚太运筹学会Pre-APORS会议代表中国运筹学会做国家贡献论文报告；2020年获中国运筹学会青年科技奖提名奖；2021年北京市青年教师教学基本功比赛二等奖。主要研究兴趣：大规模稀疏低秩优化、二阶方法、张量优化、统计学习，及其在智慧交通、压缩感知、视频分析中的应用。