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In this paper, we introduce a two-step inertial primal-dual algorithm (TSIPD) for solving the minimizations of the sum a smooth function with Lipschitzian gradient and two non-smooth convex functions with linear operators. This is a complete splitting approach, in the sense that non-smooth functions are treated separately by their proximity operators. In order to prove the convergence of the TSIPD, we transform the problem into a fixed point equation with good performance, and prove the convergence of the algorithm base on the fixed point theory. This work brings together and significantly extends several classical splitting schemes, like the primal-dual method (PD3O) proposed by Yan, and the recent three-operator splitting scheme proposed by Davis and Yin. The validity of the proposed method is demonstrated on an image denoising problem. Numerical results show that our iterative algorithm (TSIPD) has better performance than the original one (PD3O).

A Two-Step Inertial Primal-Dual Algorithm for Minimizing the Sum of Three Functions