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 - <p style="text-align: justify;">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&nbsp; numerical efficiency of our algorithms is verified by testing an image deblurring problem compared with several existing algorithms.</p>