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Yunbin Zhao
Prof. Yunbin Zhao
University of Birmingham, UK
A new computational method for the sparse point of polyhedral sets
The L0-minimization seeking the sparsest point of a polyhedral set is a long-lasting challenging problem in the fields of sparse signal recovery, statistical and machine learning, linear algebra and operations research. The weighted or reweighted L1-minimization method is one of the most plausible methods for sparsity-seeking problem. We develop a novel of such method through the fundamental complementarity theory of linear optimization. Specifically, we first prove that finding the sparse point in a polyhedron can be reformulated as the equivalent problem of seeking the corresponding densest slack variable of the dual problem of some weighted L1-problem. This theoretical observation provides a new perspective to understand the NP-hard L0-minimization, and it also provides a strong incentive to develop a new computational method, which is remarkably different from many existing sparsity-seeking methods. The weight used in our algorithm is computed via solving a convex optimization, instead of being defined directly and locally through the current iterate. Subject to suitable conditions, we rigorously show that our algorithm is globally convergent to the sparse point of the polyhedral set. The efficiency of the algorithm has been demonstrated by empirical simulations.

Yun-Bin Zhao received his BSc in Applied Mathematics from Northwestern Polytechnic University, in 1989, MSc in Applied Mathematics from Chongqing University in 1992, and PhD in Operations Research and Control Theory in 1998 from the Institute of Applied Mathematics, Chinese Academy of Sciences. From June 1998 to February 2001 he was a postdoctoral research fellow with the Institute of Computational Mathematics and Scientific/Engineering Computing, Chinese Academy of Sciences, and with the Department of SEEM, Chinese University of Hong Kong. Between 2002 and 2005, he was also a research fellow at the McMaster University and the University of Toronto in Canada. From 2001 to 2003, He was an assistant professor in the Academy of Mathematics and Systems Science (AMSS), Chinese Academy of Sciences, and from 2003 to 2007 he was an Associate Professor in AMSS. He joined the University of Birmingham in 2007 as a lecturer of theoretical and computational optimization. Since 2013, he is a Senior Lecturer in the same university. 

Yun-Bin Zhao serves as an associate editor of the international journals: Applied Mathematics and Computation (2007--), European Journal on Pure and Applied Mathematics (2008--), Journal of Algebraic Statistics (2010--2014). His research interests include the operations research, computational optimization, convex analysis, numerical linear algebra, compressed sensing, signal and image processing, and machining and statistical learning. He has published a total of more than 50 journal articles in such journals as SIAM J Optim, SIAM J Matrix Anal Appl, SIAM J Optim & Control, Math Oper Res, IEEE Trans Signal Process, J Optim Theory Appl, Appl Math Comput, Science China Math, etc. He was the principle investigator for a number of research projects funded by the National Science foundation of China (NSFC) and The Engineering and Physical Sciences Research Council (EPSRC) in UK. The purpose of his most recent research is to develop efficient computational methods for the sparse solutions of underdetermined linear system and applications to signal, image, and big-data processing, including the compressed sensing theory based on the so-called RSP tool.

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