【学术会议】On the efficient computation of a generalized Jacobian of the projector over the Birkhoff polytope

主题: On the efficient computation of a generalized Jacobian of the projector over the Birkhoff polytope

报告人: Xudong Li, Fudan University

时间: 11:00 am – 12:00 pm, April 26, 2019

地点: Boardroom, Dao Yuan Building

摘要：

In this talk, we derive an explicit formula, as well as an efficient procedure, for constructing a generalized Jacobian for the projector of a given square matrix onto the Birkhoff polytope, i.e., the set of doubly stochastic matrices. To guarantee the high efficiency of our procedure, a semismooth Newton method for solving the dual of the projection problem is proposed and efficiently implemented. Extensive numerical experiments are presented to demonstrate the merits and effectiveness of our method by comparing its performance against other powerful solvers such as the commercial software Gurobi and the academic code PPROJ [{\sc Hager and Zhang}, SIAM Journal on Optimization, 26 (2016), pp.~1773--1798]. In particular, our algorithm is able to solve the projection problem with over one billion variables and nonnegative constraints to a very high accuracy in less than 15 minutes on a modest desktop computer. More importantly, based on our efficient computation of the projections and their generalized Jacobians, we can design a highly efficient augmented Lagrangian method (ALM) for solving a class of convex quadratic programming (QP) problems constrained by the Birkhoff polytope. The resulted ALM is demonstrated to be much more efficient than Gurobi in solving a collection of QP problems arising from the relaxation of quadratic assignment problems.

简介：

Xudong Li is a tenure-track associate professor at the School of Data Science, Fudan University. He is also affiliated with the Shanghai Center for Mathematical Sciences. His research focuses on the mathematical foundation of data science with emphasis on efficient algorithms for large-scale problems arising from operations research, machine learning and statistical estimations. He received his Ph.D.in Mathematics from National University of Singapore in 2015. From 2015 to 2017, he was a research fellow in the Department of Mathematics at National University of Singapore. Prior to joining FDU in Autumn 2018, he was a postdoctoral scholar in the Department of Operations Research and Financial Engineering at Princeton University.