Recently, some attractive primal-dual algorithms have been proposed for solving a saddle-point problem, with particular applications in the area of total variation (TV) image restoration. This paper focuses on the convergence analysis of existing primal-dual algorithms and shows that the involved parameters of those primal-dual algorithms (including the step sizes) can be significantly enlarged if some simple correction steps are supplemented. As a result, we present some primal-dual-based contraction methods for solving the saddle-point problem. These contraction methods are in the prediction-correction fashion in the sense that the predictor is generated by a primal-dual method and it is corrected by some simple correction step at each iteration. In addition, based on the context of contraction type methods, we provide a novel theoretical framework for analyzing the convergence of primal-dual algorithms which simplifies existing convergence analysis substantially.
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