A Primal-Dual Fixed Point Algorithm for Multi-Block Convex Minimization

Abstract

We extend a primal-dual fixed point algorithm (PDFP) proposed in [5] 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 illustrate how practical and fully decoupled 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 present. 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 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.

Two Modified Schemes for the Primal Dual Fixed Point Method

“…The convergence analysis of PDFP was established in [7] based on a fixed point iteration. Later its three-block and multi-block extensions were studied in [8,9]. Recently it was extended to a stochastic setting for solving machine learning and image reconstruction problems in [30].…”

“…To overcome this difficulty, many algorithms based on augmented Lagrangian and Fenchel duality were designed, such as the split Bregman method [16,25] (a.k.a the alternating direction of multipliers method (ADMM) [12,19]), the primal dual hybrid gradient method (PDHG) [14,29] (also known as Chambolle-Pock algorithm [6]), Condat-Vu [11,27] algorithm, the fixed-point method based on proximity operator (FP 2 O) [22] and the primal dual fixed point method (PDFP) [7]. In this paper we focus on PDFP as it can maximally decouple subproblems and it was shown to be effective with parallel implementation for many large scale imaging and data sciences problems [7][8][9]. The scheme of PDFP for solving (1.1) is given as follows:…”