An Adaptive Prediction-Correction Method for Solving Large-Scale Nonlinear Systems of Monotone Equations with Applications

Abstract

Combining multivariate spectral gradient method with projection scheme, this paper presents an adaptive prediction-correction method for solving large-scale nonlinear systems of monotone equations. The proposed method possesses some favorable properties: (1) it is progressive step by step, that is, the distance between iterates and the solution set is decreasing monotonically; (2) global convergence result is independent of the merit function and its Lipschitz continuity; (3) it is a derivative-free method and could be applied for solving large-scale nonsmooth equations due to its lower storage requirement. Preliminary numerical results show that the proposed method is very effective. Some practical applications of the proposed method are demonstrated and tested on sparse signal reconstruction, compressed sensing, and image deconvolution problems.

1. Introduction

Considering the problem to find solutions of the following nonlinear monotone equations:𝑔(𝑥)=0, (1) where 𝑔∶ℜ^𝑛→ℜ^𝑛 is a continuous and monotone, that is, ⟨𝑔(𝑥)−𝑔(𝑦),𝑥−𝑦⟩≥0 for all 𝑥,𝑦∈ℜ^𝑛.

Nonlinear monotone equations arise in many practical applications such as ballistic trajectory computation [1] and vibration systems [2], the first-order necessary condition of the unconstrained convex optimization problem, and the subproblems in the generalized proximal algorithms with Bregman distances [3]. Moreover, we can convert some monotone variational inequality into systems of nonlinear monotone equations by means of fixed point maps or normal maps [4] if the underlying function satisfies some coercive conditions. Solodov and Svaiter [5] proposed a projection method for solving (1). A nice property of the projection method is that the whole sequence of iterates is always globally convergent to a solution of the system without any additional regularity assumptions. Moreover, Zhang and Zhou [6] presented a spectral gradient projection (SG) method for solving systems of monotone equations which combines a modified spectral gradient method and projection method. This method is shown to be globally convergent if the nonlinear monotone equations is Lipschitz continuous. Xiao et al. [7] proposed a spectral gradient method to minimize a nonsmooth minimization problem, arising from spare solution recovery in compressed sensing, consisting of a least-squares data-fitting term and a ℓ₁-norm regularization term. This problem is firstly formulated as a convex quadratic program (QP) problem and then reformulated to an equivalent nonlinear monotone equation. Furthermore, Yin et al. [8] developed a nonlinear conjugate gradient method for ℓ₁-norm regularization problems in compressed sensing. Yu [9, 10] extended the spectral gradient method and conjugate gradient-type method to solve large-scale nonlinear system of equations, respectively. Recently, the authors in [11] proposed a multivariate spectral gradient projection method for solving nonlinear monotone equations with convex constraints. Numerical results show that multivariate spectral gradient method (MSG) could improve its performance very well.

Following this line, based on multivariate spectral gradient method (MSG), we present an adaptive prediction-correction method for solving nonlinear monotone equations (1) in the next section. Its global convergence result is established, which is independent of the merit function and Lipschitz continuity. Section 3 presents some numerical experiments to demonstrate and test its practical performance on compressed sensing and image deconvolution problems. Finally, we have a conclusion section.

2. Adaptive Prediction-Correction Method

Considering the projection method [5] for solving nonlinear monotone equations (1), suppose that we have obtained a direction 𝑑_𝑘. By performing some kind of line search procedure along the direction 𝑑_𝑘, a point 𝑧_𝑘=𝑥_𝑘+𝛼_𝑘𝑑_𝑘 can be computed such that 𝑔𝑧_𝑘,𝑥_𝑘−𝑧_𝑘>0. (2) By the monotonicity of 𝑔, for any 𝑥 such that 𝑔(𝑥)=0, we have 𝑔𝑧_𝑘,𝑥−𝑧_𝑘≤0. (3) Thus, the hyperplane 𝐻_𝑘=𝑥∈ℜ^𝑛∣𝑔𝑧_𝑘,𝑥−𝑧_𝑘=0 (4) strictly separates the current iterate 𝑥_𝑘 from solutions of the systems of monotone equations. Once we get the separating hyperplane, the next iterate 𝑥_𝑘+1 is computed by projecting 𝑥_𝑘 on it.

Recalling the multivariate spectral gradient (MSG) method [12] for minimization problem min{𝑓(𝑥)∣𝑥∈ℜ^𝑛}, its iterative formula is defined by 𝑥_𝑘+1=𝑥_𝑘−diag{1/𝜆¹_𝑘,1/𝜆²_𝑘,…,1/𝜆^𝑛_𝑘}𝑔_𝑘, where 𝑔_𝑘 is the gradient of 𝑓 at 𝑥_𝑘 and diag{𝜆¹_𝑘,𝜆²_𝑘,…,𝜆^𝑛_𝑘} is obtained by minimizing diag{𝜆¹,𝜆²,…,𝜆^𝑛}𝑠_𝑘−1−𝑦_𝑘−12 (5) with respect to {𝜆^𝑖}^𝑛_𝑖=1, where 𝑠_𝑘−1=𝑥_𝑘−𝑥_𝑘−1,𝑦_𝑘=𝑔_𝑘−𝑔_𝑘−1. In particular, when 𝑓(𝑥) has positive definite diagonal Hessian matrix, multivariate spectral gradient method will be convergent quadratically [12].

Let the 𝑖th column of 𝑦_𝑘 and 𝑠_𝑘 denoted by 𝑠^𝑖_𝑘 and 𝑦^𝑖_𝑘, respectively. Combining multivariate spectral gradient method with projection scheme, we can present an adaptive prediction-correction method for solving monotone equations (1) as follows.

Algorithm 1 (multivariate spectral gradient (MSG) method). Given 𝑥₀∈ℜ^𝑛, 𝛽∈(0,1), 𝜎∈(0,1), 0<𝜀<1, 𝑟≥0, 𝛿>0. Set 𝑘=0.

Step 1. If ‖𝑔_𝑘‖=0, stop.

Step 2. (a) If 𝑘=0, set 𝑑_𝑘=−𝑔(𝑥_𝑘).
(b) else if 𝑦^𝑖_𝑘−1/𝑠^𝑖_𝑘−1>0, then set 𝜆^𝑖_𝑘=𝑦^𝑖_𝑘−1/𝑠^𝑖_𝑘−1; otherwise set 𝜆^𝑖_𝑘=(𝑠^𝑇_𝑘−1𝑦_𝑘−1)/(𝑠^𝑇_𝑘−1𝑠_𝑘−1) for 𝑖=1,2,…,𝑛, where 𝑠_𝑘−1=𝑥_𝑘−𝑥_𝑘−1, 𝑦_𝑘−1=𝑔(𝑥_𝑘)−𝑔(𝑥_𝑘−1)+𝑟𝑠_𝑘−1.
(c) else if 𝜆^𝑖_𝑘≤𝜀 or 𝜆^𝑖_𝑘≥1/𝜀, set 𝜆^𝑖_𝑘=𝛿 for 𝑖=1,2,…,𝑛.
Set 𝑑_𝑘=−diag{1/𝜆¹_𝑘,1/𝜆²_𝑘,…,1/𝜆^𝑛_𝑘}𝑔_𝑘.

Step 3 (prediction step). Compute step length 𝛼_𝑘, set 𝑧_𝑘=𝑥_𝑘+𝛼_𝑘𝑑_𝑘, where 𝛼_𝑘=𝛽^𝑚𝑘 with 𝑚_𝑘 being the smallest nonnegative integer 𝑚 such that −𝑔𝑥_𝑘+𝛽^𝑚𝑑_𝑘,𝑑_𝑘≥𝜎𝛽^𝑚𝑑_𝑘². (6)

Step 4 (correction step). Compute 𝑥_𝑘+1=𝑥_𝑘−𝑔𝑧_𝑘,𝑥_𝑘−𝑧_𝑘𝑔𝑧_𝑘²𝑔𝑧_𝑘. (7)

Step 5. Set 𝑘=𝑘+1 and go to Step 1.

By using multivariate spectral gradient method, we obtain prediction sequence {𝑧_𝑘}, and then we get correction sequence {𝑥_𝑘} via projection. It follows from (17) that 𝑥_𝑘+1 will be more close to the solution 𝑥^∗ than 𝑥_𝑘, that is, the sequence {x_𝑘} makes progress iterate by iterate. From Step 2(c), we have 1min𝜖,𝛿𝑔_𝑘≤𝑑_𝑘1≤max𝜖,1𝛿𝑔_𝑘. (8)

In what follows, we assume that 𝑔(𝑥_𝑘)≠0 for all 𝑘≥0; otherwise we have got the solution of the problem (1). The following lemma states that Algorithm 1 is well defined.

Lemma 2. There exists a nonnegative number 𝑚_𝑘 satisfying (6) for all 𝑘≥0.

Proof. Suppose that there exists a 𝑘₀≥0 such that (6) is not satisfied for any nonnegative integer 𝑚, that is, −𝑔𝑥_𝑘0+𝛽^𝑚𝑑_𝑘,𝑑_𝑘0<𝜎𝛽^𝑚𝑑_𝑘0²,∀𝑚≥1. (9) Let 𝑚→∞ and using the continuity of 𝑔 yields −𝑔𝑥_𝑘0,𝑑_𝑘0≤0. (10) From Steps 1, 2, and 5, we have 𝑔𝑥_𝑘≠0,𝑑_𝑘≠0,∀𝑘≥0. (11) Thus, −𝑔𝑥₀,𝑑₀=𝑔𝑥₀,𝑔𝑥₀−𝑔𝑥>0,_𝑘,𝑑_𝑘=𝑔𝑥_𝑘1,diag𝜆¹_𝑘,1𝜆²_𝑘1,…,𝜆^𝑛_𝑘𝑔𝑥_𝑘1≥min𝜖,𝛿𝑔_𝑘²>0,∀𝑘≥1. (12) The last inequality contradicts (10). Hence the statement is proved.

Lemma 3. Let {𝑥_𝑘} and {𝑧_𝑘} be any sequence generated by Algorithm 1. Suppose that 𝑔 is monotone and that the solution set of (1) is not empty, then {𝑥_𝑘} and {𝑧_𝑘} are both bounded. Furthermore, it holds that lim_𝑘→∞𝑥_𝑘−𝑧_𝑘=0,lim_𝑘→∞𝑥_𝑘+1−𝑥_𝑘=0. (13)(14)

Proof. From (6), we have 𝑔𝑧_𝑘,𝑥_𝑘−𝑧_𝑘=−𝛼_𝑘𝑔𝑧_𝑘,𝑑_𝑘≥𝜎𝛼²_𝑘𝑑_𝑘²=𝜎𝑥_𝑘−𝑧_𝑘². (15) Let 𝑥^∗ be an arbitrary point such that 𝑔(𝑥^∗)=0. Taking account of the monotonicity of 𝑔, we have 𝑔𝑧_𝑘,𝑥_𝑘−𝑥^∗=𝑔𝑧_𝑘,𝑥_𝑘−𝑧_𝑘+𝑔𝑧_𝑘,𝑧_𝑘−𝑥^∗≥𝑔𝑧_𝑘,𝑥_𝑘−𝑧_𝑘+𝑔𝑥^∗,𝑧_𝑘−𝑥^∗=𝑔𝑧_𝑘,𝑥_𝑘−𝑧_𝑘. (16) From (7), (14), and (16), it follows that 𝑥_𝑘+1−𝑥^∗2=𝑥_𝑘−𝑔𝑧_𝑘,𝑥_𝑘−𝑧_𝑘𝑔𝑧_𝑘²𝑔𝑧_𝑘−𝑥^∗2=𝑥_𝑘−𝑥^∗2−𝑔𝑧_𝑘,𝑥_𝑘−𝑧_𝑘²𝑔𝑧_𝑘²≤𝑥_𝑘−𝑥^∗2−𝜎²𝑥_𝑘−𝑧_𝑘⁴𝑔𝑧_𝑘². (17) Hence the sequence {‖𝑥_𝑘−𝑥^∗‖} is decreasing and convergent; moreover, the sequence {‖𝑥_𝑘‖} is bounded. Since the 𝑔 is continuous, there exists a constant 𝐶>0 such that 𝑔𝑧_𝑘≤𝐶. (18) By the Cauchy-Schwarz inequality, the monotonicity of 𝑔 and (15), we have 𝑔𝑥_𝑘≥𝑔𝑥_𝑘,𝑥_𝑘−𝑧_𝑘𝑥_𝑘−𝑧_𝑘≥𝑔𝑧_𝑘,𝑥_𝑘−𝑧_𝑘𝑥_𝑘−𝑧_𝑘≥𝜎𝑥_𝑘−𝑧_𝑘. (19) From (18) and (19), we obtain that {𝑧_𝑘} is also bounded. It follows from (17) and (18) that 𝜎²𝐶^2∞_𝑘=1𝑥_𝑘−𝑧_𝑘⁴≤^∞_𝑘=1𝑥_𝑘−𝑥^∗2−𝑥_𝑘+1−𝑥^∗2<∞, (20) which implies lim_𝑘→∞𝑥_𝑘−𝑧_𝑘=0. (21) From (7), using the Cauchy-Schwarz inequality, we obtain that 𝑥_𝑘+1−𝑥_𝑘=𝑔𝑧_𝑘,𝑥_𝑘−𝑧_𝑘𝑔𝑧_𝑘≤𝑥_𝑘−𝑧_𝑘. (22) Thus lim_𝑘→∞‖𝑥_𝑘+1−𝑥_𝑘‖=0.
The proof is complete.

Now we can establish the global convergence of Algorithm 1.

Theorem 4. Let 𝑥_𝑘 be generated by Algorithm 1; then {𝑥_𝑘} converges to an 𝑥 such that 𝑔(𝑥)=0.

Proof. Since 𝑧_𝑘=𝑥_𝑘+𝛼_𝑘𝑑_𝑘, it follows from Lemma 3 that lim_𝑘→∞𝛼_𝑘𝑑_𝑘=lim_𝑘→∞𝑥_𝑘−𝑧_𝑘=0. (23) From (8) and (18), it holds that {𝑑_𝑘} is bounded.
Now we consider the following two possible cases:(i)liminf_𝑘→∞‖𝑑(𝑥_𝑘)‖=0.(ii)liminf_𝑘→∞‖𝑑(𝑥_𝑘)‖>0.
If (i) holds, from (8), we have liminf_𝑘→∞‖𝑔(𝑥_𝑘)‖=0. By the continuity of 𝑔 and the boundedness of {𝑥_𝑘}, it is clear that the sequence {𝑥_𝑘} has some accumulation point 𝑥 such that 𝑔(𝑥)=0. From (17), we also have that the sequence {‖𝑥_𝑘−𝑥‖} converges. Therefore, {𝑥_𝑘} converges to 𝑥.
If (ii) holds, from (8), we have liminf_𝑘→∞‖𝑔(𝑥_𝑘)‖>0. By (23), it holds that lim_𝑘⟶∞𝛼_𝑘=0. (24) By the line search rule, we have for all 𝑘 sufficiently large, 𝑚_𝑘−1 will not satisfy (6). This means −𝑔𝑥_𝑘+𝛽^𝑚𝑘−1𝑑_𝑘,𝑑_𝑘<𝜎𝛽^𝑚𝑘−1𝑑_𝑘². (25) Since the sequences {𝑥_𝑘}, {𝑑_𝑘} are bounded, we choose a subsequence, let 𝑘→∞ in (25), we obtain that −𝑔𝑥^∼,𝑑^∼≤0, (26) where 𝑥^∼,𝑑^∼ are limits of corresponding subsequences. On the other hand, by (8), it holds that −𝑔𝑥^∼,𝑑^∼>0, (27) which contradicts (26). Hence, liminf_𝑘→∞‖𝑑(𝑥_𝑘)‖>0 is impossible.
The proof is complete.

3. Numerical Experiments

In this section, we report some preliminary numerical experiments to test our algorithms with comparison to spectral gradient projection method [6]. Firstly, in Section 3.1 we test these algorithms on solving nonlinear systems of monotone equations. Secondly, in Section 3.2, we apply HSG-V algorithm to solve ℓ₁-norm regularization problem arising from compressed sensing. All of numerical experiments were performed under Windows XP and MATLAB 7.0 running on a personal computer with an Intel Core 2 Duo CPU at 2.2?GHz and 2?GB of memory.

3.1. Test on Nonlinear Systems of Monotone Equations

We test the performance of our algorithms for solving some monotone equations (see details in the appendix). The termination condition is ‖𝑔(𝑥_𝑘)‖≤10⁻⁶. The parameters are specified as follows. For MSG method, we set 𝛽=0.5,𝜎=0.01,𝜖=10⁻¹⁰,𝑟=0.01. In Step 2, the parameter 𝛿 is chosen in the following way: 𝑔𝑥𝛿=1if_𝑘>1,𝑔(𝑥_𝑘)⁻¹if10⁻⁵≤𝑔𝑥_𝑘≤1,10⁵if𝑔𝑥_𝑘<10⁻⁵. (28)

Firstly, we test the performance of the MSG method on the Problem 1 with 𝑛=1000, the initial point 𝑥₀=(1,1,…,1)^𝑇. Figure 1 displays the performance of MSG method for Problem 1 which indicates that prediction sequences {𝑧_𝑘} are better than correction sequences {𝑥_𝑘} at most time. Taking this into account, we relax the MSG method such that Step 4 in Algorithm 1 is replaced by the following: ?if mod? 𝑥_𝑘+1=𝑥_𝑘−⟨𝑔(𝑧_𝑘),𝑥_𝑘−𝑧_𝑘⟩‖𝑔(𝑧_𝑘)‖²𝑔(𝑧_𝑘),?eslse? 𝑥_𝑘+1=𝑧_𝑘,?end.

(a)

(b)

Figure 1 

(a) Norm of sequence versus iteration for MSG algorithm. (b) MSG-V algorithm versus MSG and SG algorithm, where the iteration has been cut to 100 for the SG algorithm.

In this case, we refer to this modification as “MSG-V” method. When 𝑀≡1, the above algorithm will reduce to Algorithm 1. The performance of those methods on the Problem (1) is shown in Figure 1, from which we can see that the MSG-V method is preferable quite frequently to the SG method while it also outperforms the MSG method. Furthermore, motivated to accelerate the performance of MSG-V method, we present a hybrid spectral gradient (HSG-V) algorithm. The main idea of the HSG-V algorithm is to run MSG-V algorithm when 𝑦^𝑖_𝑘−1/𝑠^𝑖_𝑘−1>0 for 𝑖=1,2,…,𝑛; otherwise switch to spectral gradient projection (SG) method.

And then we compare the performance of MSG method, MSG-V method, and HSG-V method with the spectral gradient projection (SG) method in [6] on test problems with different initial points. We set 𝛽=0.5, 𝜎=0.01, 𝑟=0.01 in the spectral gradient projection (SG) method in [6], and 𝑀=10 for MSG-V method and HSG-V method.

Numerical results are shown in Tables 1, 2, 3, 4, 5, and 6 with the form NI/NF/T/BK, where we report the dimension of the problem (𝑛), the initial points (Init), the number of iteration (NI), the number of function evaluations (NF), and the CPU time (Time) in seconds and the number of backtracking (BK). The symbol “F” denotes that the method fails for this test problem, or the number of the iterations is greater than 10000.

Table 1 

Numerical results for SG/MSG methods on Problem 1.

Table 2 

Numerical results for MSG-V/HSG-V methods on Problem 1.

Table 3 

Numerical results for SG/MSG methods on Problem 2.

Table 4 

Numerical results for MSG-V/HSG-V methods on Problem 2.

Table 5 

Numerical results for SG/MSG methods on Problem 3.

Table 6 

Numerical results for MSG-V/HSG-V methods on Problem 3.

As we can see from Tables 1–6 that the HSG-V algorithm is preferable quite frequently to the SG method and also outperforms the MSG algorithm and MSG-V algorithm, since it can solve about 80% and 70% of the problems with the best time and the smallest number of function evaluations, respectively. We also find that the SG algorithm seems more sensitive to the initial points.

Figure 2 shows the performance of these algorithms relative to the number of function evaluations and CPU time, respectively, which were evaluated using the profiles of Dolan and Moré [13]. That is, for each algorithm, we plot the fraction 𝑃 of problems for which the method is within a factor 𝑡 of the smallest number of function evaluations/CPU time. Clearly, the left side of the figure gives the percentage of the test problems for which a method is the best one according to the number of function evaluations or CPU time, respectively. As we can see from Figure 2, “HSG-V” algorithm has the best performance.

(a)

(b)

Figure 2 

(a) Performance profiles for the number of function evaluations. (b) Performance profiles for the CPU time.

3.2. Test on ℓ₁-Norm Regularization Problem in Compressed Sensing

There has been considerable interest in solving the ℓ₁-norm regularized least-square problem min_{𝑥∈ℜ𝑛}𝑓(𝑥)≡12𝐴𝑥−𝑦²₂+𝜇‖𝑥‖₁, (29) where 𝐴∈ℜ^𝑚×𝑛(𝑚≪𝑛) is a linear operator, 𝑦∈ℜ^𝑚 is an observation, and 𝜇 is a nonnegative parameter. Equation (29) mainly appears in compressed sensing: an emerging methodology in digital signal processing and has attracted intensive research activities over the past few years. Compressed sensing is based on the fact that if original signal is sparse or approximately sparse in some orthogonal basis, then an exact restoration can be produced by solving (29).

Recently, Figueiredo et al. [14] proposed gradient projection method for sparse reconstruction (GPSR). The first key step of GPSR method is to express (29) as a quadratic program. For any 𝑥∈ℜ^𝑛 it can be formulated as 𝑥=𝑢−𝑣, 𝑢≥0, 𝑣≥0, where 𝑢∈ℜ^𝑛,𝑣∈ℜ^𝑛, and 𝑢_𝑖=(𝑥_𝑖)₊,𝑣_𝑖=(−𝑥_𝑖)₊ for 𝑖=1,2,…,𝑛 with (⋅)₊=max{0,⋅}. We thus have ‖𝑥‖₁=𝑒^𝑇_𝑛𝑢+𝑒^𝑇_𝑛𝑣, where 𝑒_𝑛=(1,1,…,1)^𝑇 is the vector consisting of 𝑛 ones. Hence (29) can be rewritten as the following quadratic program: min_𝑢,𝑣12𝑦−𝐴(𝑢−𝑣)²₂+𝜇𝑒^𝑇_𝑛𝑢+𝜇𝑒^𝑇_𝑛𝑣,s.t.𝑢≥0,𝑣≥0. (30) Furthermore, from [14], (30) can be written in following form min_𝑢,𝑣12𝑧^𝑇𝐵𝑧+𝑐^𝑇𝑧,s.t.𝑧≥0, (31) where 𝑢𝑣𝑧=,𝑏=𝐴^𝑇𝑦,𝑐=𝜇𝑒_2𝑛+𝑏,𝐴−𝑏𝐵=^𝑇𝐴−𝐴^𝑇𝐴−𝐴^𝑇𝐴𝐴^𝑇𝐴. (32) It is obvious that 𝐵 is a positive semidefinite matrix, hence, (30) is a convex QP problem. Figueiredo et al. [14] proposed a gradient projection method with BB step length for solving this problem.

Xiao et al. [7] indicated that the QP problem (30) is equivalent to the linear complementary problem: find 𝑧∈ℜ^2𝑛 such that 𝑧≥0,𝐵𝑧+𝑐≥0,⟨𝐵z+𝑐,𝑧⟩=0. (33) It is obvious that 𝑧 is a solution of (33) if and only if it is a solution of the following nonlinear systems of equation 𝑔(𝑧)=min{𝑧,𝐵𝑧+𝑐}=0. (34) The function 𝑔 is vector valued, and the “min” is interpreted as componentwise minimum. Xiao et al. [7] proved that 𝑔 is monotone. Hence, (34) can be solved effectively by the HSG-V algorithm.

Firstly, we consider a typical CS scenario that goal is to reconstruct a length-𝑛 sparse signal from 𝑚 observations. We measure the quality of restoration by means of squared error (MSE) to the original signal 𝑥, that is, 1MSE=𝑛𝑥−𝑥^∗2, (35) where 𝑥^∗ is the restored signal. We test a small size signal with 𝑛=2¹², 𝑚=2¹⁰, and the original contains 2⁷ randomly nonzero elements. 𝐴 is the Gaussian matrix which is generated by command 𝑟𝑎𝑛𝑑𝑛(𝑚,𝑛) in MATLAB. In this test, the measurement 𝑦 is usually contaminated by noise, that is,𝑦=𝐴𝑥+𝜔, (36) where 𝜔 is the Gaussian noise distributed as 𝑁(0,0.0001). The parameters are taken as 𝛽=0.1, 𝜎=0.01, ?𝜖=10⁻¹⁰, 𝑟=1.2, 𝑀=2, ?𝜇 is forced in decrease as the measure of [14]. To get better quality estimated signals, the process is terminated when the relative change of the objective function is below 10⁻⁵, that is, 𝑓_𝑘−𝑓_𝑘−1𝑓_𝑘−1<10⁻⁵, (37) where 𝑓_𝑘 denotes the function value at 𝑥_𝑘.

Figures 3 and 4 report the results of HSG-V for a signal sparse reconstruction from its limited measurement. Comparing the first and last plot in Figure 3, we can find that the original sparse signal is restored almost exactly from the limited measurement. From the right plot in Figure 4, we observe that all the blue dots are circled by the red circles, which shows that the original signal has been found almost exactly. All together, this simple experiment shows that HSG-V algorithms perform well, and it is an efficient method to denoise sparse signals.

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Figure 3 

(a) Original signal with length 4096 and 128 nonzero elements. (b) The noisy measurement with length 1024. (c) Recovery signal by HSG-V with 232 iterations, 15.16 s CPU time in seconds, and 3.14e-06 error.

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Figure 4 

(a) Original signal with 128 nonzero elements, (b) noisy measurement, (c) restored signal (red circles) versus original signal (blue dots) with MSE=6.72e−06, 12.66 CPU time in seconds, and 188 iterations.

In the next experiment, we compare the performance of our algorithm with the SGCS algorithm for image deconvolution, in which 𝐴 is a partial DWT matrix whose 𝑚 rows are chosen randomly from 𝑛×𝑛 DWT matrix. To measure the quality of restoration, we use the SNR (signal to noise ratio) defined as SNR=20log₁₀(‖𝑥‖̄/‖𝑥−𝑥‖̄). Figure 5 shows the original test images, and Figure 6 shows the restoration results by the SGCS and HSG-V algorithm, respectively. These results show that the HSG-V algorithm can restore blurred image quite well and obtain better quality reconstructed images in an efficient manner.

Figure 5 

The original images: cameraman/barbara/bridge.

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Figure 6 

The blurred image (first column), the restored image by SGCS algorithm (second column), and HSG-V algorithm.

4. Conclusion

In this paper, we develop an adaptive prediction-correction method for solving nonlinear monotone equations. Under some assumptions, we establish its global convergence. Base on the prediction-correction method, an efficient hybrid spectral gradient (HSG-V) algorithm is proposed, which is composite of MSG-V, algorithm and SG algorithm. Numerical results show that the HSG-V algorithm is preferable and outperforms the MSG, MSG-V and SG algorithm. Moreover, HSG-V algorithm is applied to solve ℓ₁-norm regularized problems arising from sparse signal reconstruction. Numerical experiments show that HSG-V algorithm works well, and it provides an efficient approach for compressed sensing and image deconvolution.

Appendix

The Test Problems

In this appendix, we list the test functions and the associated initial guess as follows.

Problem 1. 𝑔∶ℜ^𝑛→ℜ^𝑛, 𝑔_𝑖(𝑥)=𝑖𝑒10^𝑥𝑖−1,𝑖=1,2,…,𝑛, (A.1)𝑔(𝑥)=(𝑔₁(𝑥),𝑔₂(𝑥),…,𝑔_𝑛(𝑥))^𝑇,𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇.

Problem 2. 𝑔∶ℜ^𝑛→ℜ^𝑛, 𝑔_𝑖(𝑥)=𝑥_𝑖−sin𝑥_𝑖,𝑖=1,2,…,𝑛, (A.2)𝑔(𝑥)=(𝑔₁(𝑥),𝑔₂(𝑥),…,𝑔_𝑛(𝑥))^𝑇,𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇.

Problem 3. 𝑔∶ℜ^𝑛→ℜ^𝑛 is given by 𝑔(𝑥)=𝐴𝑥+𝑓(𝑥), (A.3)𝑓(𝑥)=(𝑒^𝑥1−1,𝑒^𝑥2−1,…,𝑒^𝑥𝑛−1)^𝑇 and 𝐴=2−1−12−1⋱⋱⋱⋱⋱−1−12. (A.4)
It is noticed that Problems 1 and 3 are smooth at 𝑥=0, while Problem 2 is nonsmooth (Table 7).

Table 7 

The initial points used in our test.

Acknowledgments

This work was partly supported by the National Natural Science Foundation of China (no. 11001060, 61262026, 81000613, 81101046), the JGZX Programme of Jiangxi Province (20112BCB23027), and the Science and Technology Programme of Jiangxi Education Committee (LDJH12088).