A new smoothing modified three-term conjugate gradient method for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$l_{1}$\end{document}l1-norm minimization problem

We consider a kind of nonsmooth optimization problems with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$l_{1}$\end{document}l1-norm minimization, which has many applications in compressed sensing, signal reconstruction, and the related engineering problems. Using smoothing approximate techniques, this kind of nonsmooth optimization problem can be transformed into a general unconstrained optimization problem, which can be solved by the proposed smoothing modified three-term conjugate gradient method. The smoothing modified three-term conjugate gradient method is based on Polak–Ribière–Polyak conjugate gradient method. For the Polak–Ribière–Polyak conjugate gradient method has good numerical properties, the proposed method possesses the sufficient descent property without any line searches, and it is also proved to be globally convergent. Finally, the numerical experiments show the efficiency of the proposed method.


Introduction
In this paper, we consider the following nonsmooth optimization problems with l 1 -norm minimization problem min x∈R n where x ∈ R n , A ∈ R m×n (m n), b ∈ R m , τ > 0, v 2 denotes the Euclidean norm of v and v 1 = n i=1 |v i | is the l 1 -norm of v. This problem is widely used in compressed sensing, signal reconstruction, analog-to-information conversion and related to many mathematical problems [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. Problem (1) is also a typical compressed sensing scenario, which can reconstruct a length-n sparse signal from m observations. From the Bayesian perspective, problem (1) can also be seen as a maximum a posteriori criterion for estimating x from observations b = Ax + ω, where ω is the Gaussian noise of variance σ 2 . Many researchers have studied the numerical algorithms, which can be used to solve problem (1) with large-scale data such as fixed point method [1], gradient projection method for sparse reconstruction [2], interior-point continuation method [3,4], iterative shrinkage thresholds algorithms in [5,6], linearized Bregman method [7,8], alternating direction algorithms [9], nonsmooth equations-based method [10] and some related methods [11,12]. Just recently, a smoothing gradient method has been given for solving problem (1) based on the new transformed absolute value equations in [14,15]. The transformation is based on the equivalence between a linear complementarity problem and an absolute value equation problem [17,18]. The complementarity problem, the absolute value equation problem, and the related constrained optimization problem are three kinds of important optimization problems [19][20][21][22][23]. On the other hand, the nonlinear conjugate gradient methods and smoothing methods are used widely to solve large-scale optimization problems [24,25], total variation image restoration [26], monotone nonlinear equations with convex constraints [27], and nonsmooth optimization problems, such as nonsmooth nonconvex problems [28], minimax problem [29], P0 nonlinear complementarity problems [30]. Specially, the effectiveness of widely used and attained different numerical outcomes three-term conjugate gradient method, which is based on Hang-Zhang conjugate gradient method and Polak-Ribière-Polyak conjugate gradient method [31][32][33], has been widely studied. Based on the above analysis, in this paper, we propose a new smoothing modified three-term conjugate gradient method to solve problem (1). The global convergence analysis of the proposed method is also presented.
The remainder of this paper is organized as follows. In Sect. 2, we give the transformation of problem (1), which includes the transformation of a linear complementarity problem transformed into an absolute value equation problem. In Sect. 3, we present the smoothing modified three-term conjugate gradient method and give the convergence analysis of it. Finally, we give some numerical results of the given method which show the effectiveness of it.

Results: the transformation of the problem
In this section, as in [9,10,14,15], we set where u i = (x i ) + and v i = (-x i ) + for all i = 1, 2, . . . , n with (x i ) + = max{x i , 0}. And we also have x 1 = 1 T n u + 1 T n v, where 1 n = [1, 1, . . . , 1] T is an n-dimensional vector with n ones. Thus, problem (1) can be transformed into the following problem: i.e., where Since H is a positive semi-definite matrix, problem (2) can be translated into a linear variable inequality problem, which is to find z ∈ R 2n such that By the feasible structure of the feasible region of z, problem (3) is equivalent to the linear complementary problem, which to find z ∈ R 2n such that Due to the equivalence of linear complementarity problems and absolute value equation problems, problem (4) can be transformed into the following absolute value equation problem, which is defined by Then problem (1) can be transformed into the following problem:

Main results and discussions
In this section, we present the smoothing modified three-term conjugate gradient method to solve problem (1). Firstly, we give the definition of smoothing function and smoothing approximation function of the absolute value function [14,15,29].
Definition 1 Let f : R n → R be a local Lipschitz continuous function. We callf : where f μ (·) is continuously differentiable in R n for any fixed μ > 0.
The smoothing function of the absolute value function is defined by and satisfies Based on (6), we obtain the following unconstrained optimization problem: (5) for i = 1, 2, . . . , 2n. Now, we give the smoothing modified three-term conjugate gradient method.
Step 2. Compute search direction by usingβ BZAU k andθ BZAU k , which are defined bỹ Step 3. Compute α k by the Armijo line search, where α k = max{ρ 0 , ρ 1 , ρ 2 , . . .} and ρ i satisfies Step 4. Compute Step 5. Set k := k + 1 and go to Step 1. Now, we give convergence analysis of Algorithm 1. In order to get the global convergence of Algorithm 1, we give the following assumptions.
(ii) There exists a positive constant L > 0 such that ∇ Zfμ (z k ) is Lipschitz continuous on an open convex set B ⊆ and for any z 1 , z 2 ∈ B, i.e., (iii) There exists a positive constant m such that where ∇ 2 zf μ (z k ) is the Hessian matrix off .
By Assumption 1, we can see that there exist positive constants γ > 0 and b such that Lemma 1 Suppose {z k } and {d k } are generated by Algorithm 1, then Proof By Algorithm 1, we have Multiplying both sides of the above equation by ∇ Zfμ (z k ) T , we obtain Hence, the proof is complete.

Lemma 2
Proof Using the techniques similar to lemmas in [31][32][33], we can get this lemma. The description will not be repeated again.
Lemma 3 Suppose Assumption 1 holds and x k and d k are generated by Algorithm 1, then where a 1 = (1σ ) -1 (m/2), m is a positive constant and 0 < σ < 1.
and τ = 6. In this example, we choose m = 100, n = 110. The numerical results are given in Figs. 5 and 6.  Example 5 In this example, we consider a typical compressed sensing problem, which is also considered in [9,10,14,15]. In this example, we choose m = 2 4 , n = 2 6 , σ = 0.5, ρ = 0.4, γ = 0.5, ε = 10 -6 , μ = 5, η = 2. The original signal contains 520 randomly generated ±1 spikes. Further, the m×n matrix A is obtained by first filling it with independent samples of a standard Gaussian distribution and then orthogonalization of its rows. In this example, we choose σ 2 = 10 -4 and τ = 0.1 A T y ∞ the same as suggested in [14]. The numerical results are shown in Fig. 9.

Conclusion
In this paper, we have proposed a new smoothing modified three-term conjugate gradient method for solving l 1 -norm nonsmooth problems. Comparing with the smoothing gradient method, GPSR method, and other methods proposed in [2,9,10,14], we can see that the smoothing modified three-term conjugate gradient method is simple and needs small storage. Comparing with the smoothing gradient method proposed in [14], the smoothing modified three-term conjugate gradient method is significantly faster especially in solving large-scale problems.