学术报告报告题目：Variational Analysis of Kurdyka-Lojasiewicz Property via Outer Limiting Subgradients
报 告 人：孟开文副教授
报告时间：2024年7月1日（星期一）16：00-17：00
报告地点：清清文理楼3A02
邀 请 人：贲树军副教授、潘少华教授
数学学院
2024年6月18日
报告摘要：The Kurdyka-Lojasiewicz (KL, for short) property, along with its exponent and modulus, has played a very important role in the study of global convergence and rate of convergence for many first-order optimization methods. In this paper, we provide a framework for the study of the KL property by way of outer limiting subgradients. For a function $f$ locally lower semicontinuous at a stationary point $\bar{x}$, we obtain some complete characterizations of the KL exponent and modulus via the outer limiting subdifferential of an auxilliary function, and provide a sufficient condition for verifying sharpness of the KL exponent. By introducing a $\frac{1}{1-\theta}$-th subderivative $h$ for $f$ at $\bar{x}$, we show that the KL property of $f$ at $\bar{x}$ with exponent $\theta\in [0, 1)$ can be inherited by $h$ at $0$ with the same exponent $\theta$, and that the KL modulus of $f$ at $\bar{x}$ is bounded above by that of $(1-\theta)h$ at $0$. When $\theta=\frac12$, we obtain the reverse results under the strong metric subregularity of the subgradient mapping for prox-regular, subdifferentially continuous and twice epi-differentiable functions, and their Moreau envelopes. The obtained results are then applied to establish the KL property with exponent $\frac12$ and to compute the corresponding KL moduli for partly smooth functions, the pointwise max of finitely many smooth functions and the $\ell_p$ ($0<p\leq 1$) regularized functions respectively.
报告人简介：孟开文，香港理工大学博士，西南财经大学数学学院副教授，博士生导师。主要从事最优化理论、算法和应用研究，主持国家自然科学基金青年和面上项目各一项。在SIAM Journal on Optimization，Operations Research， Mathematical Programming，Journal of Machine Learning Research，Journal of Global Optimization，Journal Of Convex Analysis等期刊上发表学术论文十余篇。