A Primal-Dual Fixed Point Algorithm for Multi-Block Convex Minimization (Journal of Computational Mathematics, 2021)
We have proposed a primal-dual fixed point algorithm (PDFP) for solving minimization of the sum of three convex separable functions, which involves a smooth function with Lipschitz continuous gradient, a linear composite nonsmooth function, and a nonsmooth function. Compared with similar works, the parameters in PDFP are easier to choose and are allowed in a relatively larger range. We will extend PDFP to solve two kinds of separable multi-block minimization problems, arising in signal processing and imaging science. This work shows the flexibility of applying PDFP algorithm to multi-block problems and illustrates how practical and fully splitting schemes can be derived, especially for parallel implementation of large scale problems. The connections and comparisons to the alternating direction method of multiplier (ADMM) are also presented. We demonstrate how different algorithms can be obtained by splitting the problems in different ways through the classic example of sparsity regularized least square model with constraint. In particular, for a class of linearly constrained problems, which are of great interest in the context of multi-block ADMM, can be also solved by PDFP with a guarantee of convergence. Finally, some experiments are provided to illustrate the performance of several schemes derived by the PDFP algorithm.