TY - JOUR T1 - Several Variants of the Primal-Dual Hybrid Gradient Algorithm with Applications AU - Bai , Jianchao AU - Li , Jicheng AU - Wu , Zhie JO - Numerical Mathematics: Theory, Methods and Applications VL - 1 SP - 176 EP - 199 PY - 2019 DA - 2019/12 SN - 13 DO - http://doi.org/10.4208/nmtma.OA-2019-0030 UR - https://global-sci.org/intro/article_detail/nmtma/13436.html KW - Saddle-point problem, primal-dual hybrid gradient algorithm, variational inequality, convergence complexity, image deblurring. AB -

By reviewing the primal-dual hybrid gradient algorithm (PDHG) proposed by He, You and Yuan (SIAM J. Image Sci., 7(4) (2014), pp. 2526-2537), in this paper we introduce four improved schemes for solving a class of saddle-point problems. Convergence properties of the proposed algorithms are ensured based on weak assumptions, where none of the objective functions are assumed to be strongly convex but the step-sizes in the primal-dual updates are more flexible than the previous. By making use of variational analysis, the global convergence and sublinear convergence rate in the ergodic/nonergodic sense are established, and the  numerical efficiency of our algorithms is verified by testing an image deblurring problem compared with several existing algorithms.