【12.30腾讯会议】A Lagrange-Newton Algorithm for Sparse Nonlinear Programming
报告题目：A Lagrange-Newton Algorithm for Sparse Nonlinear Programming
报告学长: Ziyan Luo（罗自炎）
报告地点：腾讯会议 ID: 381 6014 5369
摘要：The sparse nonlinear programming (SNP) problem has wide applications in signal and image processing, machine learning, pattern recognition, finance and management, etc. However, the computational challenge posed by SNP has not yet been well resolved due to the nonconvex and discontinuous -norm involved. In this paper, we resolve this numerical challenge by developing a fast Newton-type algorithm. As a theoretical cornerstone, we establish a first-order optimality condition for SNP based on the concept of strong -Lagrangian stationarity via the Lagrangian function, and reformulate it as a system of nonlinear equations called the Lagrangian equations. The nonsingularity of the corresponding Jacobian is discussed, based on which the Lagrange-Newton algorithm (LNA) is then proposed. Under mild conditions, we establish the local quadratic convergence rate and the iterative complexity estimation of LNA. To further demonstrate the efficiency and superiority of our proposed algorithm, we apply LNA to solve three specific application problems arising from compressed sensing, sparse portfolio selection and sparse principal component analysis, in which significant benefits accrue from the restricted Newton step in LNA.
This is a joint work with Chen Zhao, Naihua Xiu and Hou-Duo Qi.
报告人简介：Ziyan Luo is currently a Professor at Department of Mathematics in Beijing Jiaotong University. She received her PhD degree in Operations Research in June 2010. She was a visiting scholar with Stanford University from 2011 to 2012, and with the National University of Singapore from 2015 to 2016.
She has published over 30 research papers and has coauthored the book Tensor Analysis: Spectral Theory and Special Tensors (SIAM Press, 2017). Her research interests include sparse and low-rank optimization, statistics &optimization, tensor analysis and computation, etc.