By reviewing the primal-dual hybrid algorithm (PDHA) proposed by He, You and Yuan (SIAM J. Imaging Sci. 2014;7(4):2526-2537), in this paper we introduce four improved schemes for solving a class of generalized saddle-point problems. By making use of the variational inequality, weaker conditions are presented to ensure the global convergence of the proposed algorithms, where none of the objective functions are assumed to be strongly convex and the step-sizes in the primal-dual updates are more flexible than the previous. The global convergence and worst-case sublinear convergence rate in the ergodic/nonergodic sense are analyzed in detail. And the numerical efficiency of our algorithms is verified by testing an image deblurring problem compared with several existing algorithms.
Accepted by Numerical Mathematics Theory Methods and Applications