题目一：A Hybrid Inexact Regularized Newton and Negative Curvature Method with Complexity Guarantees for Unconstrained Nonconvex  Optimization

报告人：朱红 副教授

单  位：江苏大学

时  间：2023年6月26日 15:00

地  点：河南大学龙子湖校区九章学堂C座301

摘要：In this paper, we propose a hybrid inexact regularized Newton and negative curvature method for unconstrained nonconvex problem. We take the negative curvature or the inexact regularized direction as the descent direction based on different conditions. In addition, in order to obtain the negative curvature direction with the lowest possible computational cost, we adopt a dimensionality reduction strategy and first verify whether the Hessian matrix at the current direction in an embedded two-dimensional subspace. We show that the proposed method can find a point satisfies \(\|\nabla f(x)\|\leq \veg\) within at most \(\cO(\veg^{-3/2 - 2\delta})\) iterations for some \(\delta\in[0,1]\). Two simplified methods for nonconvex problem and strongly convex problem are analyzed as an extension of the proposed method. We show that under the local error bound of \(\|\nabla f(x)\|\), the distance between iterations generated by our proposed method and the local solution convergence to \(0\) supperlinerly, while for strongly convex problem, quadratic convergence rate can be obtained.

题目二：Iteration-complexity of a second-order augmented Lagrangian method for nonconvex unit simplex constrained optimization

报告人：朱红 副教授

单  位：江苏大学

时  间：2023年7月1日 15:00

地  点：河南大学龙子湖校区九章学堂C座301

摘要：In this paper, we give a definition of the \((\vec, \veg, \veh)\)-second-order stationary point (SOSP) for nonconvex unit simplex constrained optimization problem and present a second-order augmented Lagrangian method (SOALM) to solve it with iteration-complexity guarantee. We prove that SOALM returns an \((\vec, \veg, \veh)\)-SOSP within at most \(\widetilde{\cO}(\max\{\vec^{-1}, \veg^{-1}, \veh^{-2}\})\) outer iterations.
We give a definition of the \((\ve, \frac{1}{2})\)-SOSP for the bounded constrained AL-subproblem and propose a projected regularized Newton method with negative curvature (PRN2C) to solve it with iteration-complexity guarantee. We prove that when PRN2C is invoked in SOALM, an \((\vec, \veg, \veg^{1/2})\)-SOSP can be find within at most \(\cO(\veg^{-3/2})\) total inner iterations with high probability. Extensive numerical experiments show the effectiveness of the proposed method.

报告人简介：朱红，江苏大学副教授，硕士生导师。2016年毕业于香港浸会大学，获得哲学博士。2012年毕业于河南大学，获理学硕士。2018年12月到2019年12月受国家留学基金委的资助在加拿大西蒙弗雷泽大学访问。主要从事非线性最优化理论及非负矩阵分解等相关领域的研究。主持国家自然科学基金青年基金1项，面上项目1项，江苏省青年基金1项。