Two new Hager-Zhang iterative schemes with improved parameter choices for monotone nonlinear systems and their applications in compressed sensing

Notwithstanding its efficiency and nice attributes, most research on the iterative scheme by Hager and Zhang [Pac. J. Optim. 2(1) (2006) 35-58] are focused on unconstrained minimization problems. Inspired by this and recent works by Waziri et al. [Appl. Math. Comput. 361(2019) 645-660], Sabi’u et al. [Appl. Numer. Math. 153(2020) 217-233], and Sabi’u et al. [Int. J. Comput. Meth, doi:10.1142/S0219876220500437], this paper extends the Hager-Zhang (HZ) approach to nonlinear monotone

systems with convex constraint. Two new HZ-type iterative methods are developed by combining the prominent projection method by Solodov and Svaiter [Springer, pp 355-369, 1998] with HZ-type search directions, which are obtained by developing two new parameter choices for the Hager-Zhang scheme. The first choice, is obtained by minimizing the condition number of a modified HZ direction matrix, while the second choice is realized using singular value analysis and minimizing the

spectral condition number of the nonsingular HZ search direction matrix. Interesting properties of the schemes include solving non-smooth functions and generating descent directions. Using standard assumptions, the methods’ global convergence

are obtained and numerical experiments with recent methods in the literature, indicate that the methods proposed are promising. The schemes effectiveness are further demonstrated by their applications to sparse signal and image reconstruction problems, where they outperform some recent schemes in the literature.


INTRODUCTION
The fields of sciences, engineering, industry and other important areas of human endeavor employ vital applications with models in the form of systems of nonlinear equations. Various examples have been considered in this areas in recent years. Generally, the system of monotone nonlinear equations is formulated by: where g : R n → R n represents a continuous nonlinear mapping, which satisfies the monotonicity condition (g(x) − g(y)) T (x − y) ≥ 0, for all vectors x, y ∈ R n . (1.2) This paper focuses on the constrained version of (1.1), where the vector x lies in a nonempty closed convex set, say Ω ⊂ R n . A number of applications involve the monotone equations in (1.1) and its constrained version. For example, in radiactive transfer and transport theory [1], the popular Chandrasekhar integral equation, is decretized and presented as (1.1). In signal and image processing problems [2], monotone equations are presented as ℓ 1 − norm optimization problems. For more on the applications, see [3][4][5]. Several iterative schemes for finding solutions of monotone equations exists, and the most popular ones are the Newton and quasi-Newton schemes [6][7][8][9], which possess rapid convergence properties. These methods, however require huge matrix storage at each iterations, which makes them unpopular when engaging problems with large dimensions. Conjugate gradient (CG) method for unconstrained optimization problems is the preferable choice and most appropriate alternative to the aforementioned schemes, when dealing with problems with large dimensions. This is due to the fact that the scheme requires less memory to implement and possess strong convergence properties [10,11]. The CG iterative schemes are mostly applied to solve the following minimization problem: min x∈R n ψ(x), (1.3) with ψ : R n −→ R representing a real-valued nonlinear mapping, whose gradient is attainable. If the merit function ψ is defined by ψ := ( 1 2 )∥g(x)∥ 2 , where ∥.∥ represents the Euclidean norm, then (1.1) is clearly equivalent to the global optimization problem represented in (1.3). The CG method is implemented using the following recursive formula (1. 4) In (1.4), x k denotes the current iterate, t k a step length obtained using any proper line search procedure, and d k represents the direction given by where β k is an essential parameter, which distinguishes the CG schemes [12]. Over the decades, numerous CG algorithms conforming to various types of β k in 1.5 were presented, see [13][14][15][16][17][18][19]. Although these methods often display numerical efficiency (see [20][21][22][23][24][25][26][27][28][29][30][31][32][33]), in most cases the search directions they generate are not descent directions, namely, the condition ∇ψ(x k ) T d k ≤ −Φ∥∇ψ(x k )∥ 2 , Φ > 0, ∀k. (1.6) may not be satisfied by the methods. In order to remedy this deficiency, researchers in recent years have engaged more on CG methods with descent directions. By exploiting Perry's approach [34], Liu and Shang [35] developed a CG method with descent search directions, that yield prototypes on which other specified versions of the Perry scheme such as Hestenes-Stiefel [36] method and the Dai and Liao method [22] are developed. In [37], a new Perry CG method, which generates descent search directions irrespective of the line search used, was developed by Liu and Xu. By considering the self-scaling memoryless BFGS update, Andrei [38] proposed an adaptive class of Perry-type CG algorithms with descent search directions that are obtained as a result of symmetrization of the CG direction in [34]. Motivated by the nice properties of CG methods for unconstrained optimization, researchers have proposed their extensions for solving (1.1) and its constrained version. By combining a Polak-Ribieré-Polyak(PRP) scheme [39,40] with the projection method [41], Cheng [42] developed a CG algorithm for solving monotone nonlinear systems, which converges globally to solutions of the given problems. The authors, however indicate that global convergence of the method is dependent on its monotonicity and Lipschitz continuous property only. Also,Yu [43] presented a version of the classical PRP [39,40] method for solving nonlinear systems of equations. As with typical CG methods, the scheme is derivative-free and utilizes the line search procedure, modified by Grippo et al. [44] and Li-Fukushima [45]. Furthermore, Dai et al. [46] also proposed a derivativefree method for solving monotone nonlinear systems by combining the modified Perry method [11] with the projection method [41]. The scheme converges globally and is considered as an improvement of the classical Perry method [11]. By employing a modified secant equation and carrying out eigenvalue study of a modified Dai-Liao search direction matrix, Waziri et al. [47] proposed an effective CG method, which converges globally for nonlinear systems of equations. Using a similar approach as in [47], Waziri et al. [48] derived suitable approximations for the unmodified Hager-Zhang parameter in [49], which is considered an open problem. The new parameter choices are used to develop effective search directions, which are subsequently combined with the projection scheme [41] to develop effective methods with descent properties for solving (1.1). Sabi'u et al. [50] improved on the work in [48] by obtaining other choices for the Hager-Zhang parameter in [49], which are employed to develop other versions of the scheme in [48]. Recently, Sabi'u et al. [51] extended the Hager-Zhang scheme to solve system of monotone nonlinear equations with convex constraint, by obtaining two other choices for the Hager-Zhang parameter using singular value approach. For other recent advances in the literature, the reader is referred to [19,[53][54][55][56][57][58][59][60]. For this article, two other adaptations of HZ method, which are extension of the works in [48,50,51] are presented for solving convex constrained monotone nonlinear equations. This research is embarked upon due to the following considerations: i. The Hager-Zhang method [49] has been well-researched in unconstrained minimization, but its study in constrained monotone nonlinear systems is rare.
ii. Only few recent study of the Hager-Zhang scheme for unconstrained as well as constrained monotone system considers its applications to real life problems and situations.
Attributes of the new methods include generating descent directions, converging globally to solutions of problems and solving non-smooth functions. Apart from adding to the very few HZ methods that exist for solving systems of monotone nonlinear equations with convex constraints, another contribution of the new schemes is their applications in sparse signal reconstruction and image de-blurring problems.
We organized the remaining sections of the article as follows. Section 2 deals with preliminaries leading to derivation of the methods as well as their algorithms. Analysis of the methods' convergence properties are presented in section 3 and reports of numerical experiments conducted to highlight effectiveness of the methods proposed are presented in section 4. Applications of the methods in compressed sensing are presented in Section 5. Section 6 is used to make concluding remarks.

PRELIMINARIES AND THE PROPOSED METHODS
This section presents preliminaries as well as details involved in deriving the methods proposed. Before we proceed to drive the methods, we first describe the projection method [41], for which a sequence {z k } is generated, where and t k > 0 is a step size obtained by employing appropriate line search method in the direction d k so that g(z k ) T (x k − z k ) > 0. (2.2) Since F is a monotone mapping, then for each of solution x * of (1.1), we can have Hence, by (2.2) and (2.3) we have that the hyperplane separates the current iterate x k strictly from the solution x * of (1.1). In [41], this idea was employed and the authors suggested that the next iterate x k+1 should be the projection of x k onto the hyperplane H k and determined In [61], Zhang and Zhou presented a new line search with backtracking technique to complete the projection scheme, where the steplength t k is obtained as with γ ∈ (0, 1),σ > 0 and ζ is set as the initial value for the steplength t k . Here and in the remaining part of the article, ∥.∥ denotes the ℓ 2 − norm, ⟨., .⟩ denotes inner product of vectors, and Π Ω (.) stands for the projection operator, which exhibits the nonexpansive property: and consequently The assumptions stated below are also required in the next section of the paper: Assumption 2.1. The mapping g is monotone.
Assumption 2.3. The mapping g is Lipschitz continuous; namely, there exists a positive constant L such that for all x, y ∈ R n , the following holds: (2.10) Hager and Zhang in [62] presented a CG scheme that is much related to the method by Perry [63] and Shanno [64]. Particularly, Hager and Zhang [62,65] presented the scheme as (2.12) A remarkable feature of the method in (2.11) and (2.12) is that it generates descent directions, i.e, the condition is satisfied. In furtherance of (2.11) and (2.12) and to achieve global convergence, Hager and Zhang [62,65] improved on it and gave a modified version with update parameter andη is a positive constant. Furthermore, by making some modifications in the classical Hestenes-Stiefel (HS) [36] scheme, Hager and Zhang (HZ) [49] proposed the following one-parameter CG update It can be observed that for θ k = 2, (2.15) reduces to the scheme in (2.12). Considering that the parameter θ k is nonnegative, researchers in recent years have shown interests in finding appropriate values for the parameter. By conducting eigenvalue study Waziri et.
al. [48] modified the HZ scheme (2.15) and gave the following approximations for the parameter θ k : where R1 > 1 4 and R2 ≤ 0 and (2.18) with R3 > 1 4 and R4 ≤ 0. Clearly, in both (2.49) and (2.17) θ k > 1 4 , for which the authors showed that the scheme in [48] satisfies the inequality necessary for global convergence, namely By minimizing the Frobenius condition number of a search direction matrix as well as minimizing the difference between the smallest and largest singular value, Sabi'u et al. [50] presented two HZ schemes with the following approximations of the HZ parameter: and The authors proved the methods are effective and converge globally under mild conditions. Recently, inspired by the work in [13,48,50], Sabi'u et al. [51] proposed two other Hager-Zhang methods for convex constrained monotone systems with the following parameters: and Still, finding optimal values for θ k is an open problem for the one-parameter HZ method. So, with inspiration from the works in [48,50] and [51], we also propose two optimal choices, for the HZ parameter, and consequently present two modified HZ methods for nonlinear monotone systems with convex constraint, which are effective and always generate descent directions.

First proposed method
By considering (2.15), we suggest a modified HZ search direction defined by Proof For (2.26) to be well-defined,ỹ T k−1 s k−1 must be greater than zero. Using the definition of y k−1 , we havẽ and applying the monotonicity of the mapping g, yields Also, (2.26) can be re-written as d k = −E k g k , (2.29) where E k in (2.29) is a search direction matrix defined by Interestingly, (2.30) is similar to the quasi-Newton scheme in [18], whose updating scheme is defined as where H k approximates the inverse Hessian ∇ 2 ψ(x k ) −1 . Since E k is a rank-two update, with the formula in [18], namely we obtain its determinant as (2.33) It can be observed that the matrix in (2.31) is symmetric but the matrix E k in (2.30) is not. So, to obtain a similar form for (2.30) as (2.31), we need to symmetrize E k . Using Perry's approach [34] and employing a rank-one update, we obtain the following version of (2.30) It is observed that if we take then A k reduces to the memoryless BFGS method discussed in [52] and defined by (2.31). Using (2.34), we present the revised search direction as Now, in order to obtain other appropriate choice for the parameter θ k that will consequently guarantee descent condition as well as conditioning of the symmetric matrix A k , we conduct eigenvalue analysis of A k .

Theorem 2.2
Let the matrix A k be given by (2.34) with θ k > 1, ∀k. Then the eigenvalues of A k are 1, with multiplicity (n − 2), ξ + k and ξ − k . Also, the last two eigenvalues ξ + k and ξ − k are positive real numbers.
Proof By (2.28)ỹ T k−1 s k−1 > 0, therefore, each vectorsỹ k−1 and s k−1 are nonzero. Let ς be the vector space spanned byỹ k−1 and s k−1 , namely, ς = span{ỹ k−1 , s k−1 } ⊂ R n , then dim(ς) ≤ 2. Also, let ς ⊥ be the orthogonal complement of ς, then dim(ς ⊥ ) ≥ n − 2. Therefore, a set of mutually orthogonal vectors which from (2.34) leads to So, for i = 1, ..., n − 2, ν i k−1 , represents eigenvectors of the matrix A k corresponding to eigenvalue 1 each. Let the other two eigenvalues of A k be ξ + k and ξ − k respectively. From (2.34), we can re-write A k as which clearly represents a rank-two update. So, using similar approach as in the previous case, we obtain the determinant as (2.40) Also, from (2.34) and the fact that for a square symmetric matrix, the trace is equivalent to sum of its eigenvalues, we obtain for which we have (2.42) From (2.42), we see that for θ k > 0, Therefore, for the matrix A k to be positive-definite, we require Now, for simplicity, we set (2.46) Hence, by utilizing (2.40) and (2.42), ξ + k and ξ − k are obtained as the roots of the quadratic equation: Specifically, we can write ] . (2.48) Utilizing (2.45), (2.46), the Cauchy-Schwarz inequality and performing some algebra, we have that Now, multiplying through (2.39) by g k and using ξ − k , we have for which the positive definiteness of A k is satisfied. [66]. Given any arbitrary nonsingular matrixH, the condition number ofH is a scalar κ(H), which is defined by

Definition 2.3
In [66], it was noted that decreasing the condition number can improve the numerical stability of a matrix-based computational process. Hence, an optimal choice for θ k for which the condition number of A k is minimized, is determined by minimizing the distance between the eigenvalues ξ + k and ξ − k and making them as close as possible. Namely, the optimal value of θ k is given bȳ (2.52) Still, (2.52) as defined may fail at some instances to satisfy (2.45), in which case A k cannot be positive-definite. So, to guard against such situations, we propose the following choice for the parameter θ k : We therefore, write the revised modified HZ update parameter as with search direction given by and θ * k1 > 1 4 .
Step 2: Find z k = x k + t k d k , with t k given by (2.6) such that (2.7) is satisfied.

The second proposed method
Here, we conduct singular value study of the non-singular nonsymmetric matrix E k in (2.30) and consequently suggest another optimal value for the Hager-Zhang parameter.
First, using (2.35), we propose the following adaptive choice for θ k : where φ > 0 represents a real parameter. It can be observed that setting φ = 1, E k becomes similar to the Memoryless BFGS updating formula presented in (2.31). Next, we give the following definition: Definition 2.1. [66]. Suppose Q ∈ R n×n is a non-singular matrix having singular values Then the scalar κ(Q) is called its condition number and is defined as To proceed with our discussion, we need to determine singular values of E k . Since from for which from (2.30), we obtain This implies that the matrix E k has (n − 2) singular values with multiplicity 1. Let the remaining two singular values be τ + k and τ − k respectively. By applying properties of Frobenius norm, we have Squaring both sides of (2.68), we get (2.69) Also, given an arbitrary matrix Q ∈ R n×m with rank r, we know that (2.71) Also, when r = m = n, we have (2.73) Next, we define (2.74) Therefore, using (2.33), (2.73), (2.74) with some algebra and rearrangements, we obtain the remaining singular values of E k as Applying the Cauchy-Schwarz inequality and the fact that the parameter θ k ≥ 0, we obtain that the singular value τ + k ≥ 1. Using similar argument and the fact that τ − k ≤ τ + k , it is easily seen that τ − k ≤ 1, and we conclude that τ − k ≤ 1 ≤ τ + k . Therefore, we obtain an upper bound for the spectral condition number of E k as follows Utilizing (2.64), (2.33), and (2.73), we get (2.78) From (2.76), (2.77), and (2.78), we have . (2.79) After some algebra, we obtain the following as minimizer of the upper bound of the spectral condition number given in (2.79) Therefore, from (2.64) and (2.80), we obtain the value for the HZ parameter as But, (2.81) as defined may not lead to a descent direction in the interval (0, 1 4 ). Using earlier approach, the following revised form is proposed: We, therefore, obtain the second revised HZ update parameter as with search direction given by  . This clearly shows an improvement over the parameter choices for the schemes in [51] given by (2.23) and (2.25), both of which lie in the interval (2, ∞).
Next, we describe the algorithm as follows:

Algorithm 2: New Enhanced Hager-Zhang Projection Method (NEHZPM)
Input: Select a point Step 2: Find z k = x k + t k d k , with t k given by (2.6) such that (2.7) is satisfied.

Lemma 2.7
The line search technique (2.7) employed in step 2 of Algorithm 1 and 2 is welldefined.
Proof We approach the proof using contradiction. Let there exists an iterate indexk for which (2.7) is not satisfied. Then, by setting t¯k = ζγ i , we have By employing continuity of the function g and allowing the nonnegative integer i to approach infinity in (2.87), i.e., i → ∞, we have From (2.56) and (2.88), it is easy to deduce that g(x¯k) = 0. Interestingly, for both Algorithms 1 and 2, considering step 1 and step 2, when the line search is calculated, indicates that g(x¯k) ̸ = 0, and this clearly contradicts the result that g(x¯k) = 0.

CONVERGENCE RESULTS OF THE METHODS
Here and in the rest of this section, global convergence of Algorithms 1 and 2 is discussed. We proceed with the following Lemma, which holds for the both algorithms.
Proof Firstly, the boundedness of the sequences {x k } and {z k } has to be shown. Suppose x * ∈ Ω is a solution of (1.1). Then by monotonicity of g, we have Also from (2.9), (2.86), and the fact that 0 < Λ < 2, we have And recursively, inequality (3.4) implies that ∥x k − x * ∥ ≤ ∥x 0 − x * ∥, ∀k. Hence, the sequence {∥x k − x * ∥} is decreasing and bounded, which also implies that {x k } is bounded. Also, using Assumption 2.2, (2.10) and (3.4) we obtain Also, using (2.7) and definition of z k , we see that Utilizing the monotonicity of g and the Cauchy-Schwarz inequality, we can write Therefore, the sequence {z k } is also bounded. Moreover, the boundedness of {z k }, implies that {∥z k − x * ∥} is bounded, namely there exists τ > 0 such that for any Also, from (2.10) and (3.11), we have Also, using the line search condition (2.7), we have (3.14) By combining (3.3) and (3.14), we get Now, from (3.3) we obtain that the sequence {∥x k − x * ∥} is convergent, and {g(z k )} is also bounded by (3.13). So, taking limits of both sides of (3.15) as k grows to infinity, we obtain for which we get lim .
The subsequent theorem is used to prove global convergence of Algorithm 1.
Proof The proof takes the form of contradiction. Suppose (3.22) does not hold, then it implies there exists ε 0 > 0 such that Two cases are analyzed here: 1. If lim inf k → ∞ t k > 0, then by (2.56) and (3.17), we get lim inf k → ∞ ∥g(x k )∥ = 0, which is a contradiction with (3.23).

2.
On the other hand, if lim inf k → ∞ t k = 0, namely an infinite indexing set K exists such that lim Then from (2.7), and for Also, using Assumption 2.3, (2.56), and (2.7), we can write From both sides of the last inequality, we take limits as k grows to infinity to obtain a contradiction with (3.17), which implies that lim inf k → ∞ ∥g(x k )∥ = 0.
Since Lemma 3.1 also holds for Algorithm 2, we only need to show boundedness of the search direction generated by Algorithm 2.
We now employed the following theorem for global convergence of Algorithm 2.
The proof follows the same pattern as that of Algorithm 1. Since the directions of Algorithm 2 also satisfy (2.56), using similar arguments as for Algorithm 1, the result is obtained.

NUMERICAL EXPERIMENTS AND COMPARISONS
At this juncture, the report of a set of numerical experiments is presented to exhibit performance and efficiency of the methods proposed. We test the performances of Algorithm 1 (N I HZPM) and Algorithm 2 (NEHZPM) with the modified Hager-Zhang methods in [2,51,68] and the efficient projected gradient method in [55] all of which are called MHZM1, MHZM2, CGD and EPGM for simplicity. For all the six methods, the backtracking line search defined in (2.7) was used, and the parameters for Algorithm 1 and Algorithm 2 are set as follows: For the MHZM1, MHZM2, CGD and EPGM methods, the parameters are set as they are used by the respective authors. In addition, the codes used for implementation of all the algorithms were done in Matlab R2015a platform, and run on a personal computer with configuration (Cpu 1.8GHZ, 4GB memory). The iteration process terminates when it exceeds 1000 or ∥g( Also, reports of the experiment carried out are presented in Tables 1 − 6, where "Pnum" and "NVars" stands for problem solved and number of variables for each problem, while "Ipt" and "Inum" denotes initial guess and the total iterations achieved respectively. "Fval" and "Ptime" represent number of functions values and processing time respectively. The residual at stopping point is represented by "Norm", while " * * " indicate failure of a method to converge to a solution. The following problems were used, where g(x) = (g 1 (x), g 2 (x), ..., g n (x)) T .
Clearly, Problems 4.3, 4.4 and 4.9 are nonsmooth at x = (1, 1, ..., 1) T , x = (1, 1, ..., 1), and x = (0, 0, ..., 0) T respectively. For each of the above test functions, 24 numerical experiments were performed with variables 1000, 5000, and 10000, and the following starting points:                Furthermore, to conveniently access the impact of the choices of the HZ parameters on the new schemes, three figures are plotted by adopting the popular tool designed by Dolan and Moré [79], which can be expressed as where C represent set of experiments conducted, |C| stands for number of the problems in the set of experiments C, W denotes number of schemes considered while for each s ∈ C and w ∈ W, t s,w represents either processing time in each iteration, iterations number or function values obtained. We begin with Figure 1, which highlights performances of each of the six schemes considered in the experiments with respect to number of iterations. The figure clearly shows that the N I HZPM and NEHZPM schemes outperform the MHZM1, MHZM2, EPGM and CGD schemes by solving more problems with much less number of iterations. In order to explain this, we employ the results from Tables 1 − 6 and the summary in Table 7 Fig. 1 respectively. Next, we consider figure 2, which displays performance profile of the six methods with regards to function evaluations. The summary table and computations from Tables 1 − 6 indicate that the N I HZPM, NEHZPM and MHZM2 methods are more superior to the other three methods with respect to function evaluations as the methods solved more problems with minimum function evaluations, which can be observed from the pattern displayed by their curves in Fig. 2. As in the above case, Fig.  2 indicate that both N I HZPM and NEHZPM methods are more robust, followed by the MHZM2 scheme, than the remaining three methods, as the former's cumulative distribution function p(ϖ) attain 1 for minimal ϖ, while the latter remained below 1. Again, following similar approach, the summarized table and computations arising from results in Tables 1 − 6 involving problems, where ties are recorded by each method with at least one of the other five schemes, indicate that the MHZM1, EPGM and CGD schemes record 28.60%, 14.5%, and 20% as exhibited by their curves on the vertical axis in Fig. 2 respectively. It can be observed from the summary table that the EPGM method failed to solve any problem alone with the least function values. However, the scheme solved 14.5% of the problems with at least one of the five methods, which is clearly displayed by its curve in Fig. 2. Lastly, figure 3 shows that the N I HZPM and NEHZPM methods have an edge over the MHZM1, MHZM2, EPGM and CGD schemes, since the former solve more problems with less processing time than the latter. Careful inspection shows that the same values that are recorded in the summary table are displayed in Fig. 3. In addition, Fig. 3 shows that the N I HZPM, NEHZPM2 and MHZM2 are also more robust than the MHZM1, EPGM and CGD methods with respect to processing time. Therefore, following the above discussion and analysis, it can be concluded that the NEHZPM and N I HZPM methods are better than the MHZM1, MHZM2, EPGM and CGD methods for solving the monotone nonlinear systems of the reported collection.

THE METHODS' APPLICATIONS IN COMPRESSED SENSING
In this section, the N I HZPM and NEHZPM methods are applied to solve problems that often arise in compressed sensing; namely sparse signal reconstruction and blurred image restoration problems. Over the years, much effort have been made by researchers to obtain sparse solutions to under-determined or ill-conditioned linear systems of equations, which often arise in compressed sensing and other applications (see [80][81][82][83]). As a result, a number of iterative methods have been proposed to address the problem. The simplest approach involves minimizing a function with a quadratic ℓ 2 − norm error term and a sparseness including ℓ 1 − norm regularization term, which can be formulated as where x ∈ R n represents the signal to be reconstructed, q ∈ R l denotes an observed data, H ∈ R l×n (l ≪ n) is a linear operator, ϖ is a regularization parameter, ∥x∥ 1 and ∥x∥ 2 are the ℓ 1 and ℓ 2 norms respectively. Clearly, (5.1) represents a convex unconstrained optimization problem, which is typically found in compressed sensing. So, an original signal, which is sparse or approximately sparse can be reconstructed exactly by solving (5.1). A number of iterative methods are employed to solve (5.1) see [84][85][86][87], but the most popular are the gradient-based schemes, especially the gradient projection method for sparse reconstruction (GPSR) proposed by Figueiredo et al. [87]. Using this scheme, (5.1) is reformulated as a convex quadratic problem, where each vector x ∈ R n is split in to two parts and presented as .., n and (.) + = max{0, .}. By the ℓ 1 − norm definition, ∥x∥ 1 = E T n υ + E T n ν, where E n = (1, 1, ..., 1) T ∈ R n . Applying the above representation to (5.1), we obtain min υ,ν where F represents a vector-valued mapping. Also, since F is Lipschitz continuous and monotone (see [67,88]), it can be solved using the NIHZPM and NEHZPM schemes.

Sparse signal reconstruction with NEHZPM and EPGM methods
In this subsection, some experiments are carried out to further demonstrate performance of the NEHZPM method. The focus is on a compressive sensing problem, where the target is reconstruction of a sparse signal with length n from l observations. The quality of restoration is measured by employing the mean of square error (MSE) to the original signalx, namely withx representing the restored signal. In the experiment, we implement the NEHZPM scheme with the parameters all set as used in the earlier numerical experiments. In addition, the size of the signal is set as n = 2 12 and l = 2 10 , while the original signal contains randomly nonzero elements. Also, command randn(l, n) in Matlab generates the Gaussian matrix H. In this experiment, the measurement q is disturbed by noise, i.e q = Hx + ω, (5.8) where ω denotes the Gaussian noise distributed as N(0, 10 −4 ). In order to test the performance of the NEHZPM scheme in signal restoration, it is compared with the EPGM method [55], which was recently applied to solve the same problem. The merit function is also given as ψ(x) = 1 2 ∥q − Hx∥ 2 2 + ϖ∥x∥ 1 . As with the measure considered in [87], in this experiment the value of ϖ is forced to decrease. Also, the iterative process for the experiment is started by the measurement signal i.e, x 0 = H T q, and terminates when the relative change between successive iterates fall below 10 −5 i.e., where ψ k denotes the function value at x k . For the NEHZPM and EPGM schemes, we carried out fifteen experiments for different noise samples and report the results, which highlights the original sparse signal, the measurement as well as the reconstructed signal by each of the methods in table 8. Also, four graphs are plotted to exhibit the convergence behavior of both methods through their mean square error (MSE) and function evaluations results, as well as number of iterations and processing time respectively. As can be observed from figure 5, our proposed method exhibits much faster descent rates of MSE and function evaluations than the EPGM method. It can also be seen from the figures that the NEHZPM method requires less number of iterations and processing time in order to recover the original signal compared to that required by the EPGM method for the same process.

Image restoration experiment with NIHZPM and CGD methods
Here, some experiments are conducted with the N I HZPM and CGD methods to further highlight application and performance of the N I HZPM scheme. The experiment is on image restoration, which involves recovering or reconstructing an obscure or blurry image. Four different images are employed for the experiment, which includes, Einstein, Lena, Barbara and Cameraman. MATLAB R2014a is employed to generate all the codes with the same configuration and parameter values as used in the last experiment with γ = 0.6. Also, to obtain an insight into the performance of the N I HZPM method, it is compared with CGD [2] solver, which is used in image restoration problem. The parameters for this method are the same as used by the authors. The performance of both schemes in terms of number of iterations (Niter), processing time (PT(s)), mean square error (MSE), signal to noise ratio (SNR), which is given by wherex represents the recovered image, x denotes the original image, and the structural similarity index (SSIM), which computes the similarity between original image and the restored one in each of the experiments conducted. In the experiment, H represents a partial Discrete Wavelet Transform (DWT) matrix, for which the m rows are selected randomly from the n × n DWT matrix. The encoding matrix H is able to be tested on large images without storing any matrix, since it doesn't require storage and also enables fast matrix-vector multiplications involving H and H T . Results of the experiments conducted are presented in Table 9, while Figure 6 displays the original, blurred, and reconstructed images obtained by the N I HZPM and CGD schemes. It can be observed from Table 9 that the N I HZPM scheme has an edge over the CGD scheme in all the four metrics considered, namely objective function (ObjFun), mean square error (MSE), signal to noise ratio (SNR) and structural similarity index (SSIM). Figure 6 also show that, for the exception of Einstein, images restored from the blurred images by N I HZPM scheme appears slightly closer to the original one than the restored images by CGD method. Hence, based on this analysis, it can be concluded that the N I HZPM method is suitable for reconstruction of the images considered. The MATLAB implementation of the SSIM index can be obtained at http://www .cns .nyu .edu / lcv /ssim/.

CONCLUSION
Two effective algorithms for solving systems of nonlinear monotone equations with convex constraint are presented in this article. The schemes are based on the one-parameter Hager-Zhang method for unconstrained optimization. By carrying out eigenvalue study of a modified Hager-Zhang search direction matrix, and singular value analysis of its revised symmetric version, two new Hager-Zhang search directions are derived, which are combined with the projection technique. An attractive feature of the proposed methods is that they require low memory to implement, which is an attribute that qualifies them to solve large scaled problems. The schemes also solve nonsmooth nonlinear equations, since they are derivative-free. By applying basic conditions, we proved global convergence of the methods proposed and numerical results of experiments carried out show that both methods perform much better than the two recently proposed Hager-Zhang schemes and two other methods in the literature. To further demonstrate the effectiveness of the new methods, they are applied to solve signal and image reconstruction problems in compressed sensing. Finally, as a future research, we anticipate extending modified versions of the methods proposed to solve motion control problems in engineering and other real life applications.
Compliance with ethical standards

Conflict of Interest
The authors declare that they have no conflict of interest.