Guest Speaker: Dr. Xiaoming Yuan (Hong Kong Baptist University)

Time: 2014-12-18 (Thursday), 10：30 - 12：00
Location: Conference Room 7215, School of Science

Abstract:

      The primal-dual hybrid gradient algorithm (PDHG) has been widely used, especially for some basic image processing models. In the literature, PDHG’s convergence was established only under some restrictive conditions on its step sizes. We will revisit PDHG’s convergence in the context of a saddle-point problem, and try to better understand how to choose its step sizes. More specifically, we will show by an extremely simple example that PDHG is not necessarily convergent when the step sizes are fixed as even tiny constants. We then show that PDHG with constant step sizes is indeed convergent if one of the functions of the saddle-point problem is strong convex, a condition that does hold for some variational models in imaging. With this additional condition, we also establish a worst-case convergence rate measured by the iteration complexity for PDHG with constant step sizes. Some extensions to total variational minimization models with finite element discretization will also be mentioned.

Biography

      Xiaoming Yuan（袁晓明） was educated at Nanjing University (B.Sc., M.Phil.) and City University of Hong Kong (Ph.D.), all majoring in Mathematics. He had worked at University of Victoria, Shanghai Jiao Tong University, and University of British Columbia Okanagan before he joined Hong Kong Baptist University.
      His research focus is numerical optimization including such topics as variational inequalities and complementarity problems, sparse and low–rank optimization, and first–order methods for large–scale convex programming problems. Xiaoming Yuan is also interested in applications arising in image processing, statistics and operations management.
      His research works have appeared in some top journals such as SIAM Journal on Optimization, Mathematics of Computation, Journal of Scientific Computing and Math. Programming.