Efficient primal–dual fixed point algorithms with dynamic stepsize for composite convex optimization problems

Abstract

In this paper, we propose a new primal–dual fixed point algorithm with dynamic stepsize ($\hbox {PDFP}^{2}O_{DS_{n}}$) for solving convex minimization problems involving the sum of a smooth function with Lipschitzian gradient and the composition of a nonsmooth convex function with a continuous linear operator. Based on modified Mann iteration and the firmly nonexpansive properties of the proximity operator, we achieve the convergence of the proposed algorithm. Moreover, we give the connection of the proposed algorithm with other existing $\hbox {PDFP}^{2}O$ (Chen et al. in Inverse Probl 29:025011–025033, 2013). Finally, we illustrate the efficiency of $\hbox {PDFP}^{2}O_{DS_{n}}$ through some numerical examples on the CT image reconstruction problem. Numerical results show that our iterative algorithm ($\hbox {PDFP}^{2}O_{DS}$) performs better than the original one ($\hbox {PDFP}^{2}O$).