A partially inexact generalized primal-dual hybrid gradient method for saddle point problems with bilinear couplings

One of the most popular algorithms for saddle point problems is the so-named primal-dual hybrid gradient method, which have been received much considerable attention in the literature. Generally speaking, solving the primal and dual subproblems dominates the main computational cost of those primal-dual type methods. In this paper, we propose a partially inexact generalized primal-dual hybrid gradient method for saddle point problems with bilinear couplings, where the dual subproblem is solved approximately with a relative error strategy. Our proposed algorithm consists of two stages, where the first stage yields a predictor by solving the primal and dual subproblems, and the second procedure makes a correction on the predictor via a simple scheme. It is noteworthy that the underlying extrapolation parameter can be relaxed in a larger range, which allows us to have more choices than a fixed setting. Theoretically, we establish some convergence properties of the proposed algorithm, including the global convergence, the sub-linear convergence rate and the Q -linear convergence rate. Finally, some preliminary computational results demonstrate that our proposed algorithm works well on the fused Lasso problem with synthetic datasets and a pixel-constrained image restoration model.