Relaxed Variable Metric Primal-Dual Fixed-Point Algorithm with Applications

In this paper, a relaxed variable metric primal-dual fixed-point algorithm is proposed for solving the convex optimization problem involving the sum of two convex functions where one is differentiable with the Lipschitz continuous gradient while the other is composed of a linear operator. Based on the preconditioned forward-backward splitting algorithm, the convergence of the proposed algorithm is proved. At the same time, we show that some existing algorithms are special cases of the proposed algorithm. Furthermore, the ergodic convergence and linear convergence rates of the proposed algorithm are established under relaxed parameters. Numerical experiments on the image deblurring problems demonstrate that the proposed algorithm outperforms some existing algorithms in terms of the number of iterations.