广州数学大讲坛第一期
第七讲——哈尔滨工业大学边伟教授学术报告
题目：Sparse Optimization Problems with Cardinality Penalties: Continuous Relaxation Method and Proximal-based Algorithms
时间：2026年1月17日（星期六）下午15:30-16:30
地点：理学实验楼312
报告人：边伟 教授
摘要：Cardinality penalties represent a foundational approach for enforcing sparsity in optimization models. In this work, we propose a unified framework of continuous relaxation methods and design different proximal-friendly algorithms to four critical classes of sparse optimization problems with cardinality penalties: 1) element-wise sparse models, 2) group-structured sparse models, 3) rank-regularized models, and 4) sparse minmax models. These problems are inherently challenged by discontinuous objective functions, which complicate both analysis and computation. To bridge this gap, we first develop the exact continuous relaxation models that share the same global solutions as the original problems. A rigorous theoretical analysis is then conducted to characterize the relationships between global solutions, local solutions, and stationary points across all problem classes. Notably, we provide a more rigorous characterization to the global solutions, leading to two key definitions: (i) the strong local solutions for cardinality problems, and (ii) the strong stationary points of the continuous relaxation models, which exhibit a one-to-one correspondence. Next, leveraging this theoretical foundation, we design the proximal-based algorithms for the first three problem classes. These algorithms are provably guaranteed to converge to the strong local solutions, with some convergence rates explicitly analyzed. Some comprehensive numerical experiments further validate the efficiency of the proposed algorithms, demonstrating superior performance in both solution quality and computational scalability compared to baseline methods.
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。
