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

TL;DR

This paper introduces a primal-dual fixed-point algorithm designed to efficiently minimize the sum of three convex separable functions, applicable to image processing and signal recovery tasks with multi-regularization.

Contribution

It proposes a fully splitting symmetric scheme that involves explicit gradient and linear operators, avoiding inner iterations for certain nonsmooth functions, and proves its convergence.

Findings

Efficiently solves multi-regularization problems in image processing.

Demonstrates convergence and efficiency through fused LASSO and image restoration examples.

Handles nonsmooth functions with easily computable proximity operators.

Abstract

Many problems arising in image processing and signal recovery with multi-regularization 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 fully splitting symmetric scheme, only involving explicit gradient and linear operators without inner iteration, when the nonsmooth functions can be easily solved via their proximity operators, such as $\ell_1$ type regularization. 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.