报告题目：On solving a rank regularized minimization problem via equivalent factorized column-sparse regularized models

报 告 人：边伟 教授  哈尔滨工业大学 杰青

邀请人：赵志华  

报告时间：2025年4月22日（周二） 16:00-18:15

报告地点：北校区会议中心104

报告人简介：边伟，哈尔滨工业大学数学学院，教授、博士生导师。2004年和2009年于哈尔滨工业大学分别获得学士和博士学位。2010-2012年访问香港理工大学，跟随陈小君教授从事博士后工作。主要研究领域为：最优化理论与算法。先后在 Math. Program., Math. Oper. Res., SIAM J. Optim., SIAM J. Numer. Anal., SIAM J. Sci. Comput., SIAM J. Imaging Sci. 等期刊发表多篇学术论文。先后获国家级青年人才称号和国家级人才称号。现任SCI期刊Journal of Optimization Theory and Application编委，中国运筹学会常务理事，黑龙江省数学会常务理事，中国运筹学会数学规划分会理事等。

报告摘要：Rank regularized minimization problem is an ideal model for the low-rank matrix completion/recovery problem. The matrix factorization approach can transform the high-dimensional rank regularized problem to a low-dimensional factorized column sparse regularized problem. The latter can greatly facilitate fast computations in applicable algorithms, but needs to overcome the simultaneous non-convexity of the loss and regularization functions. In this paper, we consider the factorized column sparse regularized model. Firstly, we optimize this model with bound constraints, and establish a certain equivalence between the optimized factorization problem and rank regularized problem. Further, we strengthen the optimality condition for stationary points of the factorization problem and define the notion of strong stationary point. Moreover, we establish the equivalence between the factorization problem and its non-convex relaxation in the sense of global minimizers and strong stationary points. To solve the factorization problem, we design two types of algorithms and give an adaptive method to reduce their computation. The first algorithm is from the relaxation point of view and its iterates own some properties from global minimizers of the factorization problem after finite iterations. We give some analysis on the convergence of its iterates to a strong stationary point. The second algorithm is designed for directly solving the factorization problem. We improve the PALM algorithm introduced by Bolte et al. (Math Program Ser A 146:459–494, 2014) for the factorization problem and give its improved convergence results. Finally, we conduct numerical experiments to show the promising performance of the proposed model and algorithms for low-rank matrix completion.

Joint work with Wenjing Li and Kim-Chuan Toh

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