A primal-dual fixed point algorithm for minimization of the sum of three convex separable functions

Peijun Chen,Jianguo Huang,Xiaoqun Zhang

doi:10.1186/s13663-016-0543-2

Abstract

AbstractMany problems arising in image processing and signal recovery with multi-regularization and constraints can be formulated as minimization of a sum of three convex separable functions. Typically, the objective function involves a smooth function with Lipschitz continuous gradient, a linear composite nonsmooth function, and a nonsmooth function. In this paper, we propose a primal-dual fixed point (PDFP) scheme to solve the above class of problems. The proposed algorithm for three-block problems is a symmetric and fully splitting scheme, only involving an explicit gradient, a linear transform, and the proximity operators which may have a closed-form solution. We study the convergence of the proposed algorithm and illustrate its efficiency through examples on fused LASSO and image restoration with non-negative constraint and sparse regularization.

Highlights

In this paper, we aim to design a primal-dual fixed point algorithmic framework for solving the following minimization problem: min x∈Rn f (x) + (f ◦ B)(x) f (x), ( . )where f, f, and f are three proper lower semi-continuous convex functions, and f is differentiable on Rn with a /β-Lipschitz continuous gradient for some β ∈
For f = in ( . ), we proposed the primaldual fixed point algorithm primal-dual fixed point (PDFP) O in [ ]
6 Conclusion We have extended the algorithm preconditioned alternating projection algorithm (PAPA) [ ] and PDFP O [ ] to derive a primal-dual fixed point algorithm PDFP (see ( . )) for solving the minimization problem of three-block convex separable functions ( . )

Summary

Introduction

We aim to design a primal-dual fixed point algorithmic framework for solving the following minimization problem: min x∈Rn f (x). Chen et al Fixed Point Theory and Applications (2016) 2016:54 image restoration applications, for example in [ ] Another useful application corresponds to f = χC, where χC is the indicator function of a nonempty closed convex set C. In [ – ], several completely decoupled schemes, such as the inexact Uzawa solver and primal-dual fixed point algorithm, are proposed to avoid subproblem solving for some typical minimization problems. ). A general class of multi-step fixed point proximity algorithms is proposed in [ ], which covers several existing algorithms [ , ] as special cases. ). The convergence analysis of this PDFP algorithm is built upon fixed point theory on the primal and dual pairs. Γ is an arbitrary positive number in theory, the range of γ will affect the convergence speed and it is a difficult problem to choose a best value in practice

Linear convergence rate for special cases

Conclusion