报告题目：On some recent developments on Kurdyka-Lojasiewicz (KL) inequality

报 告 人：李国胤 教授（澳大利亚新南威尔士大学）

报告时间：2025年4月16日（星期三）10:00—11:00

报告地点：新浦金350vip一114

校内联系人：郭峰 副教授         联系方式：84708351-8602

报告摘要：The Kurdyka-Lojasiewicz (KL) inequality is a fundamental tool for analyzing the convergence of various numerical methods in solving nonsmooth and nonconvex optimization problems. In this talk, we discuss recent developments on two aspects of the KL inequality. In the first part, we establish an abstract extended convergence framework that enables one to derive superlinear convergence towards a specific target set (such as the second-order stationary points) under a generalized metric sub- regularity condition, extending the widely used analyzing framework with KL inequality. We then show that this generalized metric subreguarity for second-order stationary points can be ensured by KL inequality and the strict saddle point condition, which, in turn, is satisfied by several important applications easily. In the second part, we explain an approach for estimating the associated exponent (when it exists) in the KL inequality using a lift-and-project-approach. This enables us to estimate KL exponents for functions involving semidefinite programming representability and C^2-cone reducible structures. As an application, we establish convergence analysis for cubic regularized Newton’s method with momentum steps. Specifically, when applying this method to solve the (nonconvex) over-parameterized compressed sensing model, we obtain a (local) quadratic convergence rate to a global minimizer, under the strict complementarity condition. In the absence of the strict complementarity condition, we obtain a sublinear convergence rate of O(1/k^2) to a global minimizer. This is based on joint work with Boris Mordukhovich, Tingkei Pong, Peiran Yu and Jiangxing Zhu.

报告人简介：李国胤教授于2007年12月获香港中文大学博士学位，主要研究方向为优化理论、变分分析和多重线性代数。在澳大利亚新南威尔士大学（UNSW Sydney）完成三年博士后研究后，他于2011年受聘为该校讲师，现任数学与统计学院教授兼科研主任。李国胤教授已发表80余篇期刊论文，成果见于《Foundations of Computational Mathematics》、《SIAM Journal on Optimizationi》、《Mathematical Programming》、《Mathematics of Operations Research》、《Numerische Mathematik》、《Mathematics of Computations》及《Journal of Functional Analysis》等权威期刊。目前担任《SIAM Journal on Optimization》、《Journal of Optimization Theory and Applications》、《Optimization Letters》和《Mathematical Methods of Operations Research》编委。他于2022年获颁澳大利亚数学会奖章，并当选澳大利亚数学会会士。其研究成果屡获国际奖项，包括：《EURO Journal on Computational Optimization》Marguerite Frank最佳论文奖（2024）、国际华人数学家联盟（ICCM）最佳论文奖（2019）、《Journal of Global Optimization》最佳论文奖（2019）、《Optimization Letters》最佳论文奖（2015）。此外，他曾获澳大利亚研究理事会中期未来学者奖（2014-2018），并于2013年8月受邀任剑桥大学艾萨克·牛顿研究所访问学者。