^{1}

^{1}

^{1,2}

^{1}

^{2}

We deal with a class of problems whose objective functions are compositions of nonconvex nonsmooth functions, which has a wide range of applications in signal/image processing. We introduce a new auxiliary variable, and an efficient general proximal alternating minimization algorithm is proposed. This method solves a class of nonconvex nonsmooth problems through alternating minimization. We give a brilliant systematic analysis to guarantee the convergence of the algorithm. Simulation results and the comparison with two other existing algorithms for 1D total variation denoising validate the efficiency of the proposed approach. The algorithm does contribute to the analysis and applications of a wide class of nonconvex nonsmooth problems.

In the past few years, increasing attentions have been paid to convex optimization problems, which consist of minimizing a sum of convex or smooth functions [

Nonconvex and nonsmooth convex optimization problems are ubiquitous in different disciplines, including signal denoising [

It is quite meaningful to find a common convergent point in the optimal set of sums of simple functions [

In this application, we need to solve the following denoising problem:

Noise removal is the basis and requisite of other subsequential applications and algorithm dealing with total variation (TV) regularizer; this regularizer is of great importance since it can efficiently deal with noisy signals which have sparse derivatives (or gradients), for instance, piecewise constant (PWC) signal that has flat sections with a number of abrupt jumps. The 1D total variation minimization can be extended to related 2-dimension image restoration.

In this application, one needs to solve

In the past few years, researches on structural sparse signal recovery have been very popular and group lasso is typical one of those important problems. It attracts many attentions in face recognition, multiband signal processing, and other machine learning problems. The general case is also applied to many other kinds of structural sparse recovery problems, like

In this application, one needs to solve

The concept of deconvolution finds lots of applications in signal processing and image processing [

The main difficulty in solving (

In order to reduce computation complexity caused by composite of nonconvex function

Apparently, now this form could be solved by a series of alternating proximal minimization methods [

We then make the following assumptions about (

In fact, now the problem is a proximal alternating minimization case, whose global convergence has been detailedly analyzed in paper [

The convergence difference between algorithm of paper [

Let

For a given

When

The “limiting” subdifferential, or simply the subdifferential, of

A necessary condition for

Being given

Assume sequences

The following estimate holds:

hence

hence

For

Besides, for all bounded subsequence

Assume that

if

The above proposition gives some convergence results about sequences generated by (

This part gives more precise convergence analysis about the proximal algorithms (

Let

The function

for all

for all

If we justify that a function has K-L property, we should estimate

Below, we will give convergence analysis to critical point.

Assume that

either

or

The above theorem’s proof is based on the same analysis process in paper [

In practical scientific and engineering contexts, noise removal is the basis and requisite of other subsequential applications. It has received extensive attentions. A range of computational algorithms have been proposed to solve the denoising problem [

In 1D TV denoising problem [

In this test, we apply our algorithms (

In fact, when tests

When

Total variation denoising examples with three convex and nonconvex regularization instances (the two others are convex and nonconvex but smooth algorithms in [

Total variation denoising with nonconvex nonsmooth

According to the comparison between our algorithm for TV-

Nonconvex nonsmooth algorithm finds many interesting applications in many fields. In this paper, we give a general proximal alternating minimization method for a kind of nonconvex nonsmooth problems with complex composition. It has concise form, good theory results, and promising numerical result. For specific 1D standard TV denoising problem, the improvement is more dramatic compared to the existing algorithms [

The authors declare that they have no competing interests.

The work is supported in part by National Natural Science Foundation of China, no. 61571008.

_{0}minimization