Professor Liu gave a talk during the Computational and Applied Mathematics Colloquium. He talked about domain decomposition plays and important role in the area of numerical partial differential equation. But its value in optimization had been neglected for a long time since Jinchao Xu and his collaborator discovered it in 2002. In the era of big data, there are urgent demands for the development of distributed optimization. In this regard, we revisit the parallel subspace correction method based on domain decomposition, which was originally proposed by Xu, and propose a new algorithm framework for composite convex optimization. At each iteration, our new algorithms solve the subproblems on subspaces simultaneously to construct a search direction and take the Armijo line search to find a new iterate. Our two algorithms are called PSCLN and PSCLO, respectively, depending on whether it allows overlapping variables. The global convergence is established under mild assumptions. We compare them with state-of-the-art algorithms for solving LASSO problems, which shows that PSCLN and PSCLO often enjoy better performance. It is also observed that the overlapping scheme is helpful for the problems with structured-data.
Dr. Xin Liu got his bachelor degree at Peking University in 2004, and Phd degree at University of Chinese Academy of Sciences in 2009 under supervision of Prof. Ya-xiang Yuan (SIAM fellow, AMS fellow). He visited Zuse Institute Berlin as a post-doctor during 2009-2010 and Rice University as a visiting scholar during 2010-2011. He is now an associate professor at Academy of Mathematics and Systems Science (AMSS), Chinese Academy of Sciences, and a visiting scholar in Courant Institute of Mathematical Sciences. His research interests include numerical methods for nonlinear optimization problems, distributed optimization and numerical linear algebra. He was granted the 2016 Excellent Youth Award from the National Natural Science Foundation of China, and selected in Jingrun Chen Future Star Program of AMSS. He has served as an associate editor of Mathematical Programming Computation from 2015.