题   目：Inexact Proximal Newton Method for Nonconvex Composite Minimization

报告人：朱红 副教授

单    位：江苏大学

时    间：2025年5月27日 19:00-22:00

地    点：数学与统计学院 106研讨室

摘要：In this report, we introduce an inexact proximal Newton method for nonconvex composite problems. We establish the global convergence rate of the order \(\mathcal{O}(k^{-1/2})\) in terms of the minimal norm of the KKT residual mapping and the local superlinear convergence rate in terms of the sequence generated by the proposed algorithm under the higher-order metric \(q\)-subregularity property. When the Lipschitz constant of the corresponding gradient is known, we show that the proposed algorithm is well-defined without line search. We also introduce a stochastic block-coordinate proximal Newton method for minimizing the sum of a blockwise Lipschitz-continuously differentiable function and a separable nonsmooth convex function. In each iteration, this method randomly selects a block and approximately solves a strongly convex regularized quadratic subproblem, utilizing second-order information from the smooth component of the objective function. A backtracking line search is employed to ensure the monotonicity of the objective value. We demonstrate that under certain sampling assumption, the fundamental convergence results of our proposed stochastic method are in accordance with the corresponding results for the inexact proximal Newton method. We study the convergence of the sequence of expected objective values and the convergence of the sequence of expected residual mapping norms under various sampling assumptions. Furthermore, we introduce a method that employs the unit step size in conjunction with the Lipschitz constant of the gradient of the smooth component to formulate the strongly convex regularized quadratic subproblem. In addition to establishing the global convergence rate, we also provide a local convergence analysis for this method under certain sampling assumption and the higher-order metric subregularity of the residual mapping.

报告人简介：朱红，江苏大学副教授，硕士生导师。2016年博士毕业于香港浸会大学。主要研究方向为非线性优化及其应用。研究兴趣包括二阶算法，对偶四元数理论及应用。在 IEEE Trans. Image. Proc.,  SIAM J. Image. Sci.,  J. Sci. Comput., Inverse Prob. 等期刊发表论文20余篇，主持国家自然科学基金面上项目、青年项目、江苏省自然科学基金青年项目等。