荔园数学讲坛: Dissolving Constraints for Riemannian Optimization

主讲人：中国科学院数学与系统科学研究院 刘歆 研究员

时间：2023年3月6日9:30–10:30

地点：腾讯会议：296-245-312 密码：1234

会议链接：https://meeting.tencent.com/dm/4EOo4tMQ2xeh

内容摘要： 

In this talk, we consider optimization problems over closed embedded submanifolds of Euclidean space, which are defined by the equality constraints c(x)=0. We propose a class of constraint dissolving approaches for these Riemannian optimization problems. Its main idea is to transfer the original manifold constrained optimization into an unconstrained optimization problem which minimizes a constraint dissolving function abbreviated as CDF. Different from existing exact penalty functions, the exact gradient and Hessian of CDF are easy to compute. We study the theoretical properties of CDF and prove that the original problem and CDF share the same first-order and second-order stationary points, local minimizers, and Łojasiewicz exponents in a neighborhood of the feasible region. Remarkably, the convergence properties of our proposed constraint dissolving approaches can be directly inherited from the existing rich results in unconstrained optimization. Therefore, the proposed constraint dissolving approaches build up short cuts from unconstrained optimization to Riemannian optimization. Several illustrative examples further demonstrate the potential of our proposed constraint dissolving approaches.

主讲人简介: 

刘歆，中国科学院数学与系统科学研究院“冯康首席研究员”，博士生导师，计算数学与科学工程计算研究所副所长。

刘歆2004年本科毕业于北京大学数学科学学院；并于2009年在中国科学院数学与系统科学研究院获得博士学位；曾在德国Zuse Institute Berlin，美国Rice大学，美国纽约大学Courant研究所等科研院所长期访问。主要研究方向包括流形优化、分布式优化及其在材料计算、大数据分析和机器学习等领域的应用。刘歆于2016年获得国家优秀青年科学基金；2016年获得中国运筹学会青年科技奖；2020年获得中国工业与应用数学学会应用数学青年科技奖；2021年获得国家杰出青年科学基金。现担任Mathematical Programming Computation、Journal of Computational Mathematics、Journal of Industrial and Management Optimization等国内外期刊编委；中国运筹学会常务理事，中国工业与应用数学会副秘书长。