A fast inertial primal–dual algorithm to composite optimization models with application to image restoration problems

In this paper, we consider a fast inertial primal–dual algorithm (FIPD) for finding minimization problems of the sum of a smooth function with Lipschitzian gradient, a non-smooth proximable function, and linear composite functions. We not only prove the convergence of the proposed algorithm, but also prove that the new algorithm can achieve the worst-case o(1/k2) optimal convergence rate in terms of objective function value. This work brings together and notably extends several classical splitting schemes, like the primal–dual splitting method (PDS) proposed by Condat, the algorithm which is presented by Chambolle and Pock, as well as the recent modified fast iterative shrinkage-thresholding algorithm. The efficiency of the proposed method is demonstrated on image denoising and image deblurring. Numerical results show that our iterative algorithm (FIPD) performs better than the original one (PDS).