Open AccessArticle

An Efficient Conjugate Gradient Method for Convex Constrained Monotone Nonlinear Equations with Applications^†

KMUTTFixed Point Research Laboratory, SCL 802 Fixed Point Laboratory, Science Laboratory Building, Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand

Department of Mathematical Sciences, Faculty of Physical Sciences, Bayero University, Kano 700241, Nigeria

Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Science Laboratory Building, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand

⁴

Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan

⁵

Department of Mathematics, Faculty of Science, Gombe State University, Gombe 760214, Nigeria

Author to whom correspondence should be addressed.

^†

This project was supported by Petchra Pra Jom Klao Doctoral Academic Scholarship for Ph.D. Program at KMUTT. Moreover, this project was partially supported by the Thailand Research Fund (TRF) and the King Mongkut’s University of Technology Thonburi (KMUTT) under the TRF Research Scholar Award (Grant No. RSA6080047).

Mathematics 2019, 7(9), 767; https://doi.org/10.3390/math7090767

Submission received: 29 June 2019 / Revised: 30 July 2019 / Accepted: 6 August 2019 / Published: 21 August 2019

(This article belongs to the Special Issue Iterative Methods for Solving Nonlinear Equations and Systems)

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Browse Figures

Figure 1
Performance profiles for the number of iterations. "> Figure 2
Performance profiles for the CPU time (in seconds). "> Figure 3
Performance profiles for the number of function evaluations. "> Figure 4
Convergence histories of Algorithm 1, PCG and PDY on Problem 9. "> Figure 5
The original image (top left), the blurred image (top right), the restored image by CGD (bottom left) with SNR = 20.05, SSIM = 0.83 and by DCG (bottom right) with SNR = 20.12, SSIM = 0.83. "> Figure 6
The original image (top left), the blurred image (top right), the restored image by CGD (bottom left) with SNR = 22.93, SSIM = 0.87 and by DCG (bottom right) with SNR = 24.36, SSIM = 0.90. "> Figure 7
The original image (top left), the blurred image (top right), the restored image by CGD (bottom left) with SNR = 25.65, SSIM = 0.86 and by DCG (bottom right) with SNR = 26.37, SSIM = 0.87. "> Figure 8
The original image (top left), the blurred image (top right), the restored image by CGD (bottom left) with SNR = 21.50, SSIM = 0.84 and by DCG (bottom right) with SNR = 21.81, SSIM = 0.85. ">

Versions Notes

Abstract

This research paper proposes a derivative-free method for solving systems of nonlinear equations with closed and convex constraints, where the functions under consideration are continuous and monotone. Given an initial iterate, the process first generates a specific direction and then employs a line search strategy along the direction to calculate a new iterate. If the new iterate solves the problem, the process will stop. Otherwise, the projection of the new iterate onto the closed convex set (constraint set) determines the next iterate. In addition, the direction satisfies the sufficient descent condition and the global convergence of the method is established under suitable assumptions. Finally, some numerical experiments were presented to show the performance of the proposed method in solving nonlinear equations and its application in image recovery problems.

Keywords:

nonlinear monotone equations; conjugate gradient method; projection method; signal processing

MSC:

65K05; 90C52; 90C56; 92C55

1. Introduction

In this paper, we consider the following constrained nonlinear equation

F (x) = 0, subject to x \in Ψ,

(1)

where

F : R^{n} \to R^{n}

is continuous and monotone. The constraint set

Ψ \subset R^{n}

is nonempty, closed and convex.

Monotone equations appear in many applications [1,2,3], for example, the subproblems in the generalized proximal algorithms with Bregman distance [4], reformulation of some

ℓ_{1}

-norm regularized problems arising in compressive sensing [5] and variational inequality problems are also converted into nonlinear monotone equations via fixed point maps or normal maps [6], (see References [7,8,9] for more examples). Among earliest methods for the case

Ψ = R^{n}

is the hyperplane projection Newton method proposed by Solodov and Svaiter in Reference [10]. Subsequently, many methods were proposed by different authors. Among the popular methods are spectral gradient methods [11,12], quasi-Newton methods [13,14,15] and conjugate gradient methods (CG) [16,17].

To solve the constrained case (1), the work of Solodov and Svaiter was extended by Wang et al. [18] which also involves solving a linear system in each iteration but it was shown later by some authors that the computation of the linear system is not necessary. For examples, Xiao and Zhu [19] presented a CG method, which is a combination the well known CG-DESCENT method in Reference [20] with the projection strategy by Solodov and Svaiter. Liu et al. [21] presented two CG method with sufficiently descent directions. In Reference [22], a modified version of the method in Reference [19] was presented by Liu and Li. The modification improves the numerical performance of the method in Reference [19]. Another extension of the Dai and Kou (DK) CG method combined with the projection method to solve (1) was proposed by Ding et al. in Reference [23]. Just recently, to popularize the Dai-Yuan (DY) CG method, Liu and Feng [24] modified the DY such that the direction will be sufficiently descent. A new hybrid spectral gradient projection method for solving convex constraints nonlinear monotone equations was proposed by Awwal et al. in Reference [25]. The method is a convex combination of two different positive spectral parameters together with the projection strategy. In addition, Abubakar et al. extended the method in Reference [17] to solve (1) and also solve some sparse signal recovery problems.

Inspired by some the above methods, we propose a descent conjugate gradient method to solve problem (1). Under appropriate assumptions, the global convergence is established. Preliminary numerical experiments were given to compare the proposed method with existing methods to solve nonlinear monotone equations and some signal and image reconstruction problems arising from compressive sensing.

The remaining part of this paper is organized as follows. In Section 2, we state the proposed algorithm as well as its convergence analysis. Finally, Section 3 reports some numerical results to show the performance of the proposed method in solving Equation (1), signal recovery problems and image restoration problems.

2. Algorithm: Motivation and Convergence Result

This section starts by defining the projection map together with some of its properties.

Definition 1.

Let

Ψ \subset R^{n}

be a nonempty closed convex set. Then for any

x \in R^{n}

, its projection onto Ψ, denoted by

P_{Ψ} (x)

, is defined by

P_{Ψ} (x) = arg min {∥ x - y ∥ : y \in Ψ} .

Moreover,

P_{Ψ}

is nonexpansive, That is,

∥ P_{Ψ} (x) - P_{Ψ} (y) ∥ \leq ∥ x - y ∥, \forall x, y \in R^{n} .

(2)

All through this article, we assume the followings

( $G_{1}$ ): The mapping F is monotone, that is,

${(F (x) - F (y))}^{T} (x - y) \geq 0, \forall x, y \in R^{n} .$
( $G_{2}$ ): The mapping F is Lipschitz continuous, that is there exists a positive constant L such that

$∥ F (x) - F (y) ∥ \leq L ∥ x - y ∥, \forall x, y \in R^{n} .$
( $G_{3}$ ): The solution set of (1), denoted by $Ψ^{^{'}}$ , is nonempty.

An important property that methods for solving Equation (1) must possess is that the direction

d_{k}

satisfy

F {(x_{k})}^{T} d_{k} \leq - c {∥ F (x_{k}) ∥}^{2},

(3)

where

c > 0

is a constant. The inequality (3) is called sufficient descent property if

F (x)

is the gradient vector of a real valued function

f : R^{n} \to R

In this paper, we propose the following search direction

d_{k} = \{\begin{matrix} - F (x_{k}), & if k = 0, \\ - F (x_{k}) + β_{k} d_{k - 1} - θ_{k} F (x_{k}), & if k \geq 1, \end{matrix}

(4)

where

β_{k} = \frac{∥ F (x_{k}) ∥}{∥ d_{k - 1} ∥}

(5)

and

θ_{k}

is determined such that Equation (3) is satisfied. It is easy to see that for

k = 0

, the equation holds with

c = 1

. Now for

k \geq 1

\begin{array}{l} F {(x_{k})}^{T} d_{k} & = - F {(x_{k})}^{T} F (x_{k}) + F {(x_{k})}^{T} \frac{∥ F (x_{k}) ∥}{∥ d_{k - 1} ∥} d_{k - 1} - θ_{k} F {(x_{k})}^{T} F (x_{k}) \\ = - ∥ F (x_{k}) ∥^{2} + \frac{∥ F (x_{k}) ∥}{∥ d_{k - 1} ∥} F {(x_{k})}^{T} d_{k - 1} - θ_{k} {∥ F (x_{k}) ∥}^{2} \\ = \frac{- ∥ F (x_{k}) ∥^{2} ∥ d_{k - 1} ∥^{2} + ∥ F (x_{k}) ∥ ∥ d_{k - 1} ∥ F {(x_{k})}^{T} d_{k - 1} - θ_{k} ∥ F (x_{k}) ∥^{2} {∥ d_{k - 1} ∥}^{2}}{∥ d_{k - 1} ∥^{2}} . \end{array}

(6)

aking

θ_{k} = 1

we have

F {(x_{k})}^{T} d_{k} \leq - {∥ F (x_{k}) ∥}^{2} .

(7)

Thus, the direction defined by (4) satisfy condition (3)

\forall k

where

c = 1

To prove the global convergence of Algorithm 1, the following lemmas are needed.

Algorithm 1: (DCG)

Step 0. Given an arbitrary initial point

x_{0} \in R^{n}

, parameters

σ > 0

0 < β < 1

T o l > 0

and set

k : = 0

.
Step 1. If

∥ F (x_{k}) ∥ \leq T o l

, stop, otherwise go to Step 2.
Step 2. Compute

d_{k}

using Equation (4).
Step 3. Compute the step size

α_{k} = max {β^{i} : i = 0, 1, 2, \dots}

such that

- F {(x_{k} + α_{k} d_{k})}^{T} d_{k} \geq σ α_{k} ∥ F (x_{k} + α_{k} d_{k}) ∥ ∥ d_{k} ∥^{2} .

(8)

Step 4. Set

z_{k} = x_{k} + α_{k} d_{k}

. If

z_{k} \in Ψ

and

∥ F (z_{k}) ∥ \leq T o l

, stop. Else compute

x_{k + 1} = P_{Ψ} [x_{k} - ζ_{k} F (z_{k})]

where

ζ_{k} = \frac{F {(z_{k})}^{T} (x_{k} - z_{k})}{∥ F (z_{k}) ∥^{2}} .

Step 5. Let

k = k + 1

and go to Step 1.

Lemma 1.

The direction defined by Equation (4) satisfies the sufficient descent property, that is, there exist constants

c > 0

such that (3) holds.

Lemma 2.

Suppose that assumptions (

G_{1}

)–(

G_{3}

) holds, then the sequences

{x_{k}}

and

{z_{k}}

generated by Algorithm 1 (CGD) are bounded. Moreover, we have

lim_{k \to \infty} ∥ x_{k} - z_{k} ∥ = 0

(9)

and

lim_{k \to \infty} ∥ x_{k + 1} - x_{k} ∥ = 0 .

(10)

Proof.

We will start by showing that the sequences

{x_{k}}

and

{z_{k}}

are bounded. Suppose

\bar{x} \in Ψ^{^{'}}

, then by monotonicity of F, we get

F {(z_{k})}^{T} (x_{k} - \bar{x}) \geq F {(z_{k})}^{T} (x_{k} - z_{k}) .

(11)

Also by definition of

z_{k}

and the line search (8), we have

F {(z_{k})}^{T} (x_{k} - z_{k}) \geq σ α_{k}^{2} ∥ F (z_{k}) ∥ ∥ d_{k} ∥^{2} \geq 0 .

(12)

So, we have

\begin{array}{l} ∥ x_{k + 1} - \bar{x} ∥^{2} & = ∥ P_{Ψ} [x_{k} - ζ_{k} F (z_{k})] - \bar{x} ∥^{2} \leq {∥ x_{k} - ζ_{k} F (z_{k}) - \bar{x} ∥}^{2} \\ = ∥ x_{k} - \bar{x} ∥^{2} - 2 ζ_{k} F {(z_{k})}^{T} (x_{k} - \bar{x}) + {∥ ζ F (z_{k}) ∥}^{2} \\ \leq ∥ x_{k} - \bar{x} ∥^{2} - 2 ζ_{k} F {(z_{k})}^{T} (x_{k} - z_{k}) + {∥ ζ F (z_{k}) ∥}^{2} \\ = ∥ x_{k} - \bar{x} ∥^{2} - {(\frac{F {(z_{k})}^{T} (x_{k} - z_{k})}{∥ F (z_{k}) ∥})}^{2} \\ \leq ∥ x_{k} - \bar{x} ∥^{2} \end{array}

(13)

Thus the sequence

{∥ x_{k} - \bar{x} ∥}

is non increasing and convergent and hence

{x_{k}}

is bounded. Furthermore, from Equation (13), we have

∥ x_{k + 1} - \bar{x} ∥^{2} \leq {∥ x_{k} - \bar{x} ∥}^{2},

(14)

and we can deduce recursively that

∥ x_{k} - \bar{x} ∥^{2} \leq {∥ x_{0} - \bar{x} ∥}^{2}, \forall k \geq 0 .

Then from Assumption (

G_{2}

), we obtain

∥ F (x_{k}) ∥ = ∥ F (x_{k}) - F (\bar{x}) ∥ \leq L ∥ x_{k} - \bar{x} ∥ \leq L ∥ x_{0} - \bar{x} ∥ .

If we let

L ∥ x_{0} - \bar{x} ∥ = κ

, then the sequence

{F (x_{k})}

is bounded, that is,

∥ F (x_{k}) ∥ \leq κ, \forall k \geq 0 .

(15)

By the definition of

z_{k}

, Equation (12), monotonicity of F and the Cauchy-Schwatz inequality, we get

σ ∥ x_{k} - z_{k} ∥ = \frac{σ ∥ α_{k} d_{k} ∥^{2}}{∥ x_{k} - z_{k} ∥} \leq \frac{F {(z_{k})}^{T} (x_{k} - z_{k})}{∥ x_{k} - z_{k} ∥} \leq \frac{F {(z_{k})}^{T} (x_{k} - z_{k})}{∥ x_{k} - z_{k} ∥} \leq ∥ F (x_{k}) ∥ .

(16)

The boundedness of the sequence

{x_{k}}

together with Equations (15) and (16), implies the sequence

{z_{k}}

is bounded.

Since

{z_{k}}

is bounded, then for any

\bar{x} \in Ψ

, the sequence

{z_{k} - \bar{x}}

is also bounded, that is, there exists a positive constant

ν > 0

such that

∥ z_{k} - \bar{x} ∥ \leq ν .

This together with Assumption (

G_{2}

) yields

∥ F (z_{k}) ∥ = ∥ F (z_{k}) - F (\bar{x}) ∥ \leq L ∥ z_{k} - \bar{x} ∥ \leq L ν .

Therefore, using Equation (13), we have

\frac{σ^{2}}{{(L ν)}^{2}} ∥ x_{k} - z_{k} ∥^{4} \leq ∥ x_{k} - \bar{x} ∥^{2} - {∥ x_{k + 1} - \bar{x} ∥}^{2},

which implies

\frac{σ^{2}}{{(L ν)}^{2}} \sum_{k = 0}^{\infty} ∥ x_{k} - z_{k} ∥^{4} \leq \sum_{k = 0}^{\infty} (∥ x_{k} - \bar{x} ∥^{2} - ∥ x_{k + 1} - \bar{x} ∥^{2}) \leq ∥ x_{0} - \bar{x} ∥ < \infty .

(17)

Equation (17) implies

lim_{k \to \infty} ∥ x_{k} - z_{k} ∥ = 0 .

However, using Equation (2), the definition of

ζ_{k}

and the Cauchy-Schwartz inequality, we have

\begin{array}{l} ∥ x_{k + 1} - x_{k} ∥ & = ∥ P_{Ψ} [x_{k} - ζ_{k} F (z_{k})] - x_{k} ∥ \\ \leq ∥ x_{k} - ζ_{k} F (z_{k}) - x_{k} ∥ \\ = ∥ ζ_{k} F (z_{k}) ∥ \\ = ∥ x_{k} - z_{k} ∥, \end{array}

(18)

which yields

lim_{k \to \infty} ∥ x_{k + 1} - x_{k} ∥ = 0 .

☐

Equation (9) and definition of

z_{k}

implies that

lim_{k \to \infty} α_{k} ∥ d_{k} ∥ = 0 .

(19)

Lemma 3.

Suppose

d_{k}

is generated by Algorithm 1 (CGD), then there exist

M > 0

such the

∥ d_{k} ∥ \leq M

Proof.

By definition of

d_{k}

and Equation (15)

\begin{array}{l} ∥ d_{k} ∥ & = ∥ - 2 F (x_{k}) + \frac{∥ F (x_{k}) ∥}{∥ d_{k - 1} ∥} d_{k - 1} ∥ \\ \leq 2 ∥ F (x_{k}) ∥ + \frac{∥ F (x_{k}) ∥}{∥ d_{k - 1} ∥} ∥ d_{k - 1} ∥ \\ \leq 3 ∥ F (x_{k}) ∥ \\ \leq 3 κ . \end{array}

(20)

Letting

M = 3 κ

, we have the desired result. ☐

Theorem 1.

Suppose that assumptions (

G_{1}

)–(

G_{3}

) hold and let the sequence

{x_{k}}

be generated by Algorithm 1, then

\underset{k \to \infty}{lim inf} ∥ F (x_{k}) ∥ = 0,

(21)

Proof.

To prove the Theorem, we consider two cases;

Case 1

Suppose

\underset{k \to \infty}{lim inf} ∥ d_{k} ∥ = 0,

we have

\underset{k \to \infty}{lim inf} ∥ F (x_{k}) ∥ = 0 .

Then by continuity of F, the sequence

{x_{k}}

has some accumulation point

\bar{x}

such that

F (\bar{x}) = 0 .

Because

{∥ x_{k} - \bar{x} ∥}

converges and

\bar{x}

is an accumulation point of

{x_{k}}

, therefore

{x_{k}}

converges to

\bar{x}

Case 2

Suppose

\underset{k \to \infty}{lim inf} ∥ d_{k} ∥ > 0,

we have

\underset{k \to \infty}{lim inf} ∥ F (x_{k}) ∥ > 0 .

Then by (19), it holds that

lim_{k \to \infty} α_{k} = 0 .

Also from Equation (8),

- F {(x_{k} + β^{i - 1} d_{k})}^{T} d_{k} < σ β^{i - 1} ∥ F (x_{k} + β^{i - 1} d_{k}) ∥ ∥ d_{k} ∥^{2}

and the boundedness of

{x_{k}}, {d_{k}}

, we can choose a sub-sequence such that allowing k to go to infinity in the above inequality results

F {(\bar{x})}^{T} \bar{d} > 0 .

(22)

On the other hand, allowing k to approach ∞ in (7), implies

F {(\bar{x})}^{T} \bar{d} \leq 0 .

(23)

(22) and (23) imply contradiction. Hence,

\underset{k \to \infty}{lim inf} ∥ F (x_{k}) ∥ > 0

is not true and the proof is complete. ☐

3. Numerical Examples

This section gives the performance of the proposed method with existing methods such as PCG and PDY proposed in References [22,24], respectively, to solve monotone nonlinear equations using 9 benchmark test problems. Furthermore Algorithm 1 is applied to restore a blurred image. All codes were written in MATLAB R2018b and run on a PC with intel COREi5 processor with 4 GB of RAM and CPU 2.3 GHZ. All runs were stopped whenever

∥ F (x_{k}) ∥ < 10^{- 5} .

The parameters chosen for the existing algorithm are as follows:

PCG method: All parameters are chosen as in Reference [22].

PDY method: All parameters are chosen as in Reference [24].

Algorithm 1: We have tested several values of

β \in (0, 1)

and found that

β = 0.7

gives the best result. In addition, to implement most of the optimization algorithms, the parameter

σ

is chosen as a very small number. Therefore, we chose

β = 0.7

and

σ = 0.0001

for the implementation of the proposed algorithm.

We test 9 different problems with dimensions ranging from

n = 1000, 5000, 10, 000, 50, 000, 100, 000

and 6 initial points:

x_{1} = {(0.1, 0.1, \dots, 1)}^{T}

x_{2} = {(0.2, 0.2, \dots, 0.2)}^{T}

x_{3} = {(0.5, 0.5, \dots, 0.5)}^{T}

x_{4} = {(1.2, 1.2, \dots, 1.2)}^{T}

x_{5} = {(1.5, 1.5, \dots, 1.5)}^{T}

x_{6} = {(2, 2, \dots, 2)}^{T}

. In Table 1, Table 2, Table 3, Table 4, Table 5, Table 6, Table 7, Table 8 and Table 9, the number of iterations (ITER), number of function evaluations (FVAL), CPU time in seconds (TIME) and the norm at the approximate solution (NORM) were reported. The symbol ‘−’ is used when the number of iterations exceeds 1000 and/or the number of function evaluations exceeds 2000.

The test problems are listed below, where the function F is taken as

F (x) = {(f_{1} (x), f_{2} (x), \dots, f_{n} (x))}^{T}

Problem 1

([26]). Exponential Function.

\begin{array}{l} f_{1} (x) & = e^{x_{1}} - 1, \\ f_{i} (x) & = e^{x_{i}} + x_{i} - 1, for i = 2, 3, \dots, n, \\ a n d & Ψ = R_{+}^{n} . \end{array}

Problem 2

([26]). Modified Logarithmic Function.

\begin{array}{l} f_{i} (x) & = ln (x_{i} + 1) - \frac{x_{i}}{n}, for i = 2, 3, \dots, n, \\ a n d & Ψ = {x \in R^{n} : \sum_{i = 1}^{n} x_{i} \leq n, x_{i} > - 1, i = 1, 2, \dots, n} . \end{array}

Problem 3

([13]). Nonsmooth Function.

\begin{array}{l} f_{i} (x) & = 2 x_{i} - sin | x_{i} |, i = 1, 2, 3, \dots, n, \\ a n d Ψ & = {x \in R^{n} : \sum_{i = 1}^{n} x_{i} \leq n, x_{i} \geq 0, i = 1, 2, \dots, n} . \end{array}

It is clear that Problem 3 is nonsmooth at

x = 0

Problem 4

([26]). Strictly Convex Function I.

\begin{array}{l} f_{i} (x) & = e^{x_{i}} - 1, for i = 1, 2, \dots, n, \\ a n d & Ψ = R_{+}^{n} . \end{array}

Problem 5

([26]). Strictly Convex Function II.

\begin{array}{l} f_{i} (x) & = \frac{i}{n} e^{x_{i}} - 1, for i = 1, 2, \dots, n, \\ a n d & Ψ = R_{+}^{n} . \end{array}

Problem 6

([27]). Tridiagonal Exponential Function

\begin{array}{l} f_{1} (x) & = x_{1} - e^{cos (h (x_{1} + x_{2}))}, \\ f_{i} (x) & = x_{i} - e^{cos (h (x_{i - 1} + x_{i} + x_{i + 1}))}, for i = 2, \dots, n - 1, \\ f_{n} (x) & = x_{n} - e^{cos (h (x_{n - 1} + x_{n}))}, \\ h & = \frac{1}{n + 1} a n d Ψ = R_{+}^{n} . \end{array}

Problem 7

([28]). Nonsmooth Function

\begin{array}{l} f_{i} (x) & = x_{i} - sin | x_{i} - 1 |, i = 1, 2, 3, \dots, n . \\ a n d Ψ & = {x \in R^{n} : \sum_{i = 1}^{n} x_{i} \leq n, x_{i} \geq - 1, i = 1, 2, \dots, n} . \end{array}

Problem 8

([23]). Penalty 1

\begin{array}{l} t_{i} & = \sum_{i = 1}^{n} x_{i}^{2}, c = 10^{- 5} \\ f_{i} (x) & = 2 c (x_{i} - 1) + 4 (t_{i} - 0.25) x_{i}, i = 1, 2, 3, \dots, n . \\ a n d Ψ & = R_{+}^{n} . \end{array}

Problem 9

([29]). Semismooth Function

\begin{array}{l} f_{1} (x) & = x_{1} + x_{1}^{3} - 10, \\ f_{2} (x) & = x_{2} - x_{3} + x_{2}^{3} + 1, \\ f_{3} (x) & = x_{2} + x_{3} + 2 x_{3}^{3} - 3, \\ f_{4} (x) & = 2 x_{4}^{3}, \\ a n d Ψ & = {x \in R^{4} : \sum_{i = 1}^{4} x_{i} \leq 3, x_{i} \geq 0, i = 1, 2, 3, 4} . \end{array}

In addition, we employ the performance profile developed in Reference [30] to obtain Figure 1, Figure 2 and Figure 3, which is a helpful process of standardizing the comparison of methods. The measure of the performance profile considered are; number of iterations, CPU time (in seconds) and number of function evaluations. Figure 1 reveals that Algorithm 1 most performs better in terms of number of iterations, as it solves and wins 90 percent of the problems with less number of iterations, while PCG and PDY solves and wins less than 10 percent. In Figure 2, Algorithm 1 performed a little less by solving and winning over 80 percent of the problems with less CPU time as against PCG and PDY with similar performance of less than 10 percent of the problems considered. The translation of Figure 3 is identical to Figure 1. Figure 4 is the plot of the decrease in residual norm against number of iterations on problem 9 with

x_{4}

as initial point. It shows the speed of the convergence of each algorithm using the convergence tolerance

10^{- 5}

, it can be observed that Algorithm 1 converges faster than PCG and PDY.

Applications in Compressive Sensing

There are many problems in signal processing and statistical inference involving finding sparse solutions to ill-conditioned linear systems of equations. Among popular approach is minimizing an objective function which contains quadratic (

ℓ_{2}

) error term and a sparse

ℓ_{1}

—regularization term, that is,

min_{x} \frac{1}{2} {∥ y - B x ∥}_{2}^{2} + η {∥ x ∥}_{1},

(24)

where

x \in R^{n}

y \in R^{k}

is an observation,

B \in R^{k \times n}

(

k < < n

) is a linear operator,

η

is a non-negative parameter,

{∥ x ∥}_{2}

denotes the Euclidean norm of x and

{∥ x ∥}_{1} = \sum_{i = 1}^{n} | x_{i} |

is the

ℓ_{1}

—norm of x. It is easy to see that problem (24) is a convex unconstrained minimization problem. Due to the fact that if the original signal is sparse or approximately sparse in some orthogonal basis, problem (24) frequently appears in compressive sensing and hence an exact restoration can be produced by solving (24).

Iterative methods for solving (24) have been presented in many papers (see References [5,31,32,33,34,35]). The most popular method among these methods is the gradient based method and the earliest gradient projection method for sparse reconstruction (GPRS) was proposed by Figueiredo et al. [5]. The first step of the GPRS method is to express (24) as a quadratic problem using the following process. Let

x \in R^{n}

and splitting it into its positive and negative parts. Then x can be formulated as

x = u - v, u \geq 0, v \geq 0,

where

u_{i} = {(x_{i})}_{+}

v_{i} = {(- x_{i})}_{+}

for all

i = 1, 2, \dots, n

and

{(.)}_{+} = max {0, .}

. By definition of

ℓ_{1}

-norm, we have

{∥ x ∥}_{1} = e_{n}^{T} u + e_{n}^{T} v

, where

e_{n} = {(1, 1, \dots, 1)}^{T} \in R^{n}

. Now (24) can be written as

min_{u, v} \frac{1}{2} {∥ y - B (u - v) ∥}_{2}^{2} + η e_{n}^{T} u + η e_{n}^{T} v, u \geq 0, v \geq 0,

(25)

which is a bound-constrained quadratic program. However, from Reference [5], Equation (25) can be written in standard form as

min_{z} \frac{1}{2} z^{T} D z + c^{T} z, such that z \geq 0,

(26)

where

z = (\begin{matrix} u \\ v \end{matrix})

c = ω e_{2 n} + (\begin{matrix} - b \\ b \end{matrix})

b = B^{T} y

D = (\begin{matrix} B^{T} B & - B^{T} B \\ - B^{T} B & B^{T} B \end{matrix})

Clearly, D is a positive semi-definite matrix, which implies that Equation (26) is a convex quadratic problem.

Xiao et al. [19] translated (26) into a linear variable inequality problem which is equivalent to a linear complementarity problem. Furthermore, it was noted that z is a solution of the linear complementarity problem if and only if it is a solution of the nonlinear equation:

F (z) = min {z, D z + c} = 0 .

(27)

The function F is a vector-valued function and the “min” is interpreted as component-wise minimum. It was proved in References [36,37] that

F (z)

is continuous and monotone. Therefore problem (24) can be translated into problem (1) and thus Algorithm 1 (DCG) can be applied to solve it.

In this experiment, we consider a simple compressive sensing possible situation, where our goal is to restore a blurred image. We use the following well-known gray test images; (P1) Cameraman, (P2) Lena, (P3) House and (P4) Peppers for the experiments. We use 4 different Gaussian blur kernals with standard deviation

σ

to compare the robustness of DCG method with CGD method proposed in Reference [19]. CGD method is an extension of the well-known conjugate gradient method for unconstrained optimization CG-DESCENT [20] to solve the

ℓ_{1}

-norm regularized problems.

To access the performance of each algorithm tested with respect to metrics that indicate a better quality of restoration, in Table 10 we reported the number of iterations, the objective function (ObjFun) value at the approximate solution, the mean of squared error (MSE) to the original image

\tilde{x}

M S E = \frac{1}{n} {∥ \tilde{x} - x_{*} ∥}^{2},

where

x_{*}

is the reconstructed image and the signal-to-noise-ratio (SNR) which is defined as

SNR = 20 \times {log}_{10} (\frac{∥ \bar{x} ∥}{∥ x - \bar{x} ∥}) .

We also reported the structural similarity (SSIM) index that measure the similarity between the original image and the restored image [38]. The MATLAB implementation of the SSIM index can be obtained at http://www.cns.nyu.edu/~lcv/ssim/.

The original, blurred and restored images by each of the algorithm are given in Figure 5, Figure 6, Figure 7 and Figure 8. The figures demonstrate that both the two tested algorithm can restored the blurred images. It can be observed from Table 10 and Figure 5, Figure 6, Figure 7 and Figure 8 that Algorithm 1 (DCG) compete with the CGD algorithm, therefore, it can be used as an alternative to CGD for restoring blurred image.

4. Conclusions

In this research article, we present a CG method which possesses the sufficient descent property for solving constrained nonlinear monotone equations. The proposed method has the ability to solve non-smooth equations as it does not require matrix storage and Jacobian information of the nonlinear equation under consideration. The sequence of iterates generated converge the solution under appropriate assumptions. Finally, we give some numerical examples to display the efficiency of the proposed method in terms of number of iterations, CPU time and number of function evaluations compared with some related methods for solving convex constrained nonlinear monotone equations and its application in image restoration problems.

Author Contributions

Conceptualization, A.B.A.; methodology, A.B.A.; software, H.M.; validation, P.K. and A.M.A.; formal analysis, P.K. and H.M.; investigation, P.K. and A.M.A.; resources, P.K.; data curation, A.B.A. and H.M.; writing—original draft preparation, A.B.A.; writing—review and editing, H.M.; visualization, A.M.A.; supervision, P.K.; project administration, P.K.; funding acquisition, P.K.

Funding

Petchra Pra Jom Klao Doctoral Scholarship for Ph.D. program of King Mongkut’s University of Technology Thonburi (KMUTT) and Theoretical and Computational Science (TaCS) Center. Moreover, this project was partially supported by the Thailand Research Fund (TRF) and the King Mongkut’s University of Technology Thonburi (KMUTT) under the TRF Research Scholar Award (Grant No. RSA6080047).

Acknowledgments

We thank Associate Professor Jin Kiu Liu for providing us with the access of the CGD-CS MATLAB codes. The authors acknowledge the financial support provided by King Mongkut’s University of Technology Thonburi through the “KMUTT 55th Anniversary Commemorative Fund”. This project is supported by the theoretical and computational science (TaCS) center under computational and applied science for smart research innovation (CLASSIC), Faculty of Science, KMUTT. The first author was supported by the “Petchra Pra Jom Klao Ph.D. Research Scholarship from King Mongkut’s University of Technology Thonburi”.

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. Performance profiles for the number of iterations.

Figure 2. Performance profiles for the CPU time (in seconds).

Figure 3. Performance profiles for the number of function evaluations.

Figure 4. Convergence histories of Algorithm 1, PCG and PDY on Problem 9.

Figure 5. The original image (top left), the blurred image (top right), the restored image by CGD (bottom left) with SNR = 20.05, SSIM = 0.83 and by DCG (bottom right) with SNR = 20.12, SSIM = 0.83.

Figure 6. The original image (top left), the blurred image (top right), the restored image by CGD (bottom left) with SNR = 22.93, SSIM = 0.87 and by DCG (bottom right) with SNR = 24.36, SSIM = 0.90.

Figure 7. The original image (top left), the blurred image (top right), the restored image by CGD (bottom left) with SNR = 25.65, SSIM = 0.86 and by DCG (bottom right) with SNR = 26.37, SSIM = 0.87.

Figure 8. The original image (top left), the blurred image (top right), the restored image by CGD (bottom left) with SNR = 21.50, SSIM = 0.84 and by DCG (bottom right) with SNR = 21.81, SSIM = 0.85.

Table 1. Numerical Results for Algorithm 1 (DCG), PCG and PDY for Problem 1 with given initial points and dimensions.

		Algorithm 1				PCG				PDY
DIMENSION	INITIAL POINT	ITER	FVAL	TIME	NORM	ITER	FVAL	TIME	NORM	ITER	FVAL	TIME	NORM
1000	$x_{1}$	11	49	0.025557	8.88 $\times 10^{- 6}$	18	73	0.019295	5.72 $\times 10^{- 6}$	12	49	0.16248	9.18 $\times 10^{- 6}$
	$x_{2}$	12	53	0.014164	4.78 $\times 10^{- 6}$	18	73	0.011648	9.82 $\times 10^{- 6}$	13	53	0.03780	6.35 $\times 10^{- 6}$
	$x_{3}$	12	53	0.008524	8.75 $\times 10^{- 6}$	19	77	0.011197	7.1 $\times 10^{- 6}$	14	57	0.01550	5.59 $\times 10^{- 6}$
	$x_{4}$	13	57	0.011333	6.68 $\times 10^{- 6}$	18	73	0.022197	8.27 $\times 10^{- 6}$	15	61	0.01746	4.07 $\times 10^{- 6}$
	$x_{5}$	13	57	0.014202	6.09 $\times 10^{- 6}$	63	254	0.046072	9.58 $\times 10^{- 6}$	14	57	0.02193	9.91 $\times 10^{- 6}$
	$x_{6}$	13	57	0.011045	8.14 $\times 10^{- 6}$	61	246	0.031608	9.15 $\times 10^{- 6}$	40	162	0.03472	9.70 $\times 10^{- 6}$
5000	$x_{1}$	12	53	0.024311	5.82 $\times 10^{- 6}$	18	73	0.11431	7.42 $\times 10^{- 6}$	13	53	0.03158	6.87 $\times 10^{- 6}$
	$x_{2}$	13	57	0.027361	3.13 $\times 10^{- 6}$	19	77	0.03997	6.53 $\times 10^{- 6}$	14	57	0.04270	4.62 $\times 10^{- 6}$
	$x_{3}$	13	57	0.02541	5.73 $\times 10^{- 6}$	20	81	0.056159	5.2 $\times 10^{- 6}$	15	61	0.05433	4.18 $\times 10^{- 6}$
	$x_{4}$	14	61	0.032038	4.38 $\times 10^{- 6}$	19	77	0.038381	8.1 $\times 10^{- 6}$	15	61	0.04357	9.08 $\times 10^{- 6}$
	$x_{5}$	14	61	0.039044	3.98 $\times 10^{- 6}$	62	250	0.15836	9.53 $\times 10^{- 6}$	15	61	0.08960	7.30 $\times 10^{- 6}$
	$x_{6}$	14	61	0.027231	5.33 $\times 10^{- 6}$	60	242	0.13276	9.1 $\times 10^{- 6}$	39	158	0.11284	9.86 $\times 10^{- 6}$
10,000	$x_{1}$	12	53	0.05434	8.23 $\times 10^{- 6}$	18	73	0.073207	9.5 $\times 10^{- 6}$	13	53	0.06371	9.70 $\times 10^{- 6}$
	$x_{2}$	13	57	0.045664	4.43 $\times 10^{- 6}$	19	77	0.090771	8.15 $\times 10^{- 6}$	14	57	0.06336	6.53 $\times 10^{- 6}$
	$x_{3}$	13	57	0.041922	8.09 $\times 10^{- 6}$	20	81	0.070859	6.74 $\times 10^{- 6}$	15	61	0.06414	5.90 $\times 10^{- 6}$
	$x_{4}$	14	61	0.047641	6.2 $\times 10^{- 6}$	20	81	0.087357	5.11 $\times 10^{- 6}$	16	65	0.07920	4.28 $\times 10^{- 6}$
	$x_{5}$	14	61	0.045734	5.62 $\times 10^{- 6}$	62	250	0.24646	8.87 $\times 10^{- 6}$	39	158	0.22101	7.97 $\times 10^{- 6}$
	$x_{6}$	14	61	0.057104	7.54 $\times 10^{- 6}$	59	238	0.19949	9.96 $\times 10^{- 6}$	87	351	0.36237	9.93 $\times 10^{- 6}$
50,000	$x_{1}$	13	57	0.16384	5.41 $\times 10^{- 6}$	19	77	0.25487	8.8 $\times 10^{- 6}$	14	57	0.27607	7.12 $\times 10^{- 6}$
	$x_{2}$	13	57	0.18633	9.9 $\times 10^{- 6}$	20	81	0.32689	7.39 $\times 10^{- 6}$	15	61	0.26220	4.91 $\times 10^{- 6}$
	$x_{3}$	14	61	0.20801	5.32 $\times 10^{- 6}$	21	85	0.33649	6.31 $\times 10^{- 6}$	16	65	0.28260	4.37 $\times 10^{- 6}$
	$x_{4}$	15	65	0.1946	4.08 $\times 10^{- 6}$	21	85	0.32779	5.1 $\times 10^{- 6}$	38	154	0.60650	7.54 $\times 10^{- 6}$
	$x_{5}$	15	65	0.19799	3.69 $\times 10^{- 6}$	61	246	0.82615	8.85 $\times 10^{- 6}$	177	712	2.52330	9.44 $\times 10^{- 6}$
	$x_{6}$	15	65	0.22418	4.95 $\times 10^{- 6}$	59	238	0.79992	8.5 $\times 10^{- 6}$	361	1449	5.97950	9.74 $\times 10^{- 6}$
100,000	$x_{1}$	13	57	0.32291	7.65 $\times 10^{- 6}$	20	81	0.53846	5.52 $\times 10^{- 6}$	15	61	0.39342	3.39 $\times 10^{- 6}$
	$x_{2}$	14	61	0.33329	4.12 $\times 10^{- 6}$	21	85	0.61533	4.62 $\times 10^{- 6}$	15	61	0.42154	6.94 $\times 10^{- 6}$
	$x_{3}$	14	61	0.37048	7.52 $\times 10^{- 6}$	21	85	0.53638	8.78 $\times 10^{- 6}$	16	65	0.45851	6.18 $\times 10^{- 6}$
	$x_{4}$	15	65	0.36058	5.76 $\times 10^{- 6}$	21	85	0.62002	7.21 $\times 10^{- 6}$	175	704	4.36100	9.47 $\times 10^{- 6}$
	$x_{5}$	15	65	0.34975	5.22 $\times 10^{- 6}$	60	242	1.4564	9.73 $\times 10^{- 6}$	176	708	4.29180	9.91 $\times 10^{- 6}$
	$x_{6}$	15	65	0.3621	7.01 $\times 10^{- 6}$	58	234	1.4155	9.42 $\times 10^{- 6}$	360	1445	9.71190	9.99 $\times 10^{- 6}$

Table 2. Numerical Results for Algorithm 1 (DCG), PCG and PDY for Problem 2 with given initial points and dimensions.

		Algorithm 1				PCG				PDY
DIMENSION	INITIAL POINT	ITER	FVAL	TIME	NORM	ITER	FVAL	TIME	NORM	ITER	FVAL	TIME	NORM
1000	$x_{1}$	9	38	3.1744	5.84 $\times 10^{- 6}$	15	59	0.049899	8.59 $\times 10^{- 6}$	10	39	0.01053	6.96 $\times 10^{- 6}$
	$x_{2}$	10	42	0.014633	6.25 $\times 10^{- 6}$	11	42	0.015089	9.07 $\times 10^{- 6}$	11	43	0.00937	9.23 $\times 10^{- 6}$
	$x_{3}$	9	38	0.017067	7.4 $\times 10^{- 6}$	17	66	0.016935	6.44 $\times 10^{- 6}$	13	51	0.01111	6.26 $\times 10^{- 6}$
	$x_{4}$	7	30	0.006392	6.53 $\times 10^{- 6}$	18	69	0.01436	6 $\times 10^{- 6}$	14	55	0.02154	9.46 $\times 10^{- 6}$
	$x_{5}$	11	46	0.011954	3.47 $\times 10^{- 6}$	13	48	0.00907	7.58 $\times 10^{- 6}$	15	59	0.01850	4.60 $\times 10^{- 6}$
	$x_{6}$	12	50	0.68666	6.74 $\times 10^{- 6}$	18	68	0.01352	5.4 $\times 10^{- 6}$	15	59	0.01938	7.71 $\times 10^{- 6}$
5000	$x_{1}$	10	42	0.11241	3.53 $\times 10^{- 6}$	16	63	0.041151	9.35 $\times 10^{- 6}$	11	43	0.03528	4.86 $\times 10^{- 6}$
	$x_{2}$	11	46	0.028723	3.81 $\times 10^{- 6}$	12	46	0.028706	8.8 $\times 10^{- 6}$	12	47	0.04032	6.89 $\times 10^{- 6}$
	$x_{3}$	10	42	0.029367	4.3 $\times 10^{- 6}$	18	70	0.047532	6.98 $\times 10^{- 6}$	14	55	0.04889	4.61 $\times 10^{- 6}$
	$x_{4}$	13	54	0.036231	3.67 $\times 10^{- 6}$	19	73	0.052164	6.45 $\times 10^{- 6}$	15	59	0.04826	6.96 $\times 10^{- 6}$
	$x_{5}$	11	46	0.04963	7.21 $\times 10^{- 6}$	14	52	0.040529	6.71 $\times 10^{- 6}$	16	63	0.05969	3.37 $\times 10^{- 6}$
	$x_{6}$	13	54	0.054971	4.05 $\times 10^{- 6}$	19	72	0.12303	5.71 $\times 10^{- 6}$	16	63	0.06253	5.64 $\times 10^{- 6}$
10,000	$x_{1}$	10	42	0.049614	4.98 $\times 10^{- 6}$	17	67	0.074779	6.6 $\times 10^{- 6}$	11	43	0.06732	6.85 $\times 10^{- 6}$
	$x_{2}$	11	46	0.061595	5.36 $\times 10^{- 6}$	13	50	0.08308	6.11 $\times 10^{- 6}$	12	47	0.12232	9.72 $\times 10^{- 6}$
	$x_{3}$	10	42	0.054587	6.02 $\times 10^{- 6}$	18	70	0.085554	9.83 $\times 10^{- 6}$	14	55	0.08288	6.51 $\times 10^{- 6}$
	$x_{4}$	13	54	0.073333	5.16 $\times 10^{- 6}$	19	73	0.10579	9.07 $\times 10^{- 6}$	15	59	0.08413	9.82 $\times 10^{- 6}$
	$x_{5}$	12	50	0.06306	2.83 $\times 10^{- 6}$	14	52	0.074982	9.18 $\times 10^{- 6}$	16	63	0.09589	4.75 $\times 10^{- 6}$
	$x_{6}$	13	54	0.062259	5.69 $\times 10^{- 6}$	19	72	0.099167	8.02 $\times 10^{- 6}$	16	64	0.11499	8.55 $\times 10^{- 6}$
50,000	$x_{1}$	11	46	0.20703	3.1 $\times 10^{- 6}$	18	71	0.39473	7.37 $\times 10^{- 6}$	12	47	0.27826	5.23 $\times 10^{- 6}$
	$x_{2}$	12	50	0.23251	3.35 $\times 10^{- 6}$	14	54	0.27346	6.74 $\times 10^{- 6}$	13	51	0.29642	7.11 $\times 10^{- 6}$
	$x_{3}$	11	46	0.21338	3.73 $\times 10^{- 6}$	20	78	0.37249	5.5 $\times 10^{- 6}$	15	59	0.35602	4.82 $\times 10^{- 6}$
	$x_{4}$	14	58	0.3232	3.22 $\times 10^{- 6}$	21	81	0.37591	5.07 $\times 10^{- 6}$	35	141	0.69470	6.69 $\times 10^{- 6}$
	$x_{5}$	12	50	0.22703	6.27 $\times 10^{- 6}$	16	60	0.26339	5.02 $\times 10^{- 6}$	35	141	0.68488	9.12 $\times 10^{- 6}$
	$x_{6}$	14	58	0.25979	3.54 $\times 10^{- 6}$	20	76	0.33814	8.93 $\times 10^{- 6}$	35	141	0.70973	9.91 $\times 10^{- 6}$
100,000	$x_{1}$	11	46	0.55511	4.38 $\times 10^{- 6}$	19	75	0.65494	5.22 $\times 10^{- 6}$	12	47	0.44541	7.39 $\times 10^{- 6}$
	$x_{2}$	12	50	0.54694	4.73 $\times 10^{- 6}$	14	54	0.4944	9.52 $\times 10^{- 6}$	14	55	0.53299	3.39 $\times 10^{- 6}$
	$x_{3}$	11	46	0.40922	5.27 $\times 10^{- 6}$	20	78	0.78319	7.78 $\times 10^{- 6}$	15	60	0.58603	8.71 $\times 10^{- 6}$
	$x_{4}$	14	58	0.62049	4.55 $\times 10^{- 6}$	21	81	0.76051	7.17 $\times 10^{- 6}$	72	290	2.70630	8.31 $\times 10^{- 6}$
	$x_{5}$	12	50	0.47039	8.86 $\times 10^{- 6}$	16	60	0.58545	7.07 $\times 10^{- 6}$	72	290	2.72220	8.68 $\times 10^{- 6}$
	$x_{6}$	14	58	0.71174	5.01 $\times 10^{- 6}$	21	80	0.77051	6.32 $\times 10^{- 6}$	72	290	2.75850	8.96 $\times 10^{- 6}$

Table 3. Numerical Results for Algorithm 1 (DCG), PCG and PDY for Problem 3 with given initial points and dimensions.

		Algorithm 1 (DCG)				PCG				PDY
DIMENSION	INITIAL POINT	ITER	FVAL	TIME	NORM	ITER	FVAL	TIME	NORM	ITER	FVAL	TIME	NORM
1000	$x_{1}$	10	43	0.75322	9.9 $\times 10^{- 6}$	19	76	0.55752	5.62 $\times 10^{- 6}$	12	48	0.01255	4.45 $\times 10^{- 6}$
	$x_{2}$	11	47	0.006933	5.46 $\times 10^{- 6}$	20	80	0.010936	5.58 $\times 10^{- 6}$	12	48	0.01311	9.02 $\times 10^{- 6}$
	$x_{3}$	12	51	0.00676	3.48 $\times 10^{- 6}$	21	84	0.011048	6.58 $\times 10^{- 6}$	13	52	0.01486	8.34 $\times 10^{- 6}$
	$x_{4}$	12	51	0.009664	4.41 $\times 10^{- 6}$	22	88	0.011058	5.67 $\times 10^{- 6}$	14	56	0.01698	8.04 $\times 10^{- 6}$
	$x_{5}$	11	47	0.010487	9.06 $\times 10^{- 6}$	22	88	0.012198	5.64 $\times 10^{- 6}$	14	56	0.01551	9.72 $\times 10^{- 6}$
	$x_{6}$	13	55	0.012702	3.15 $\times 10^{- 6}$	21	84	0.018231	8.36 $\times 10^{- 6}$	14	56	0.01534	9.42 $\times 10^{- 6}$
5000	$x_{1}$	11	47	0.019458	6.19 $\times 10^{- 6}$	20	80	0.040808	6.29 $\times 10^{- 6}$	12	48	0.03660	9.94 $\times 10^{- 6}$
	$x_{2}$	12	51	0.021562	3.42 $\times 10^{- 6}$	21	84	0.06688	6.25 $\times 10^{- 6}$	13	52	0.03616	6.85 $\times 10^{- 6}$
	$x_{3}$	12	51	0.024274	7.79 $\times 10^{- 6}$	22	88	0.04144	7.37 $\times 10^{- 6}$	14	56	0.04594	6.14 $\times 10^{- 6}$
	$x_{4}$	12	51	0.026771	9.86 $\times 10^{- 6}$	23	92	0.052214	6.35 $\times 10^{- 6}$	15	60	0.04342	6.01 $\times 10^{- 6}$
	$x_{5}$	12	51	0.026814	5.67 $\times 10^{- 6}$	23	92	0.041444	6.31 $\times 10^{- 6}$	15	60	0.04296	7.25 $\times 10^{- 6}$
	$x_{6}$	13	55	0.023903	7.03 $\times 10^{- 6}$	22	88	0.040135	9.37 $\times 10^{- 6}$	32	129	0.10081	8.85 $\times 10^{- 6}$
10,000	$x_{1}$	11	47	0.044134	8.75 $\times 10^{- 6}$	20	80	0.064312	8.9 $\times 10^{- 6}$	13	52	0.06192	4.77 $\times 10^{- 6}$
	$x_{2}$	12	51	0.051947	4.83 $\times 10^{- 6}$	21	84	0.088102	8.84 $\times 10^{- 6}$	13	52	0.06442	9.68 $\times 10^{- 6}$
	$x_{3}$	13	55	0.057291	3.08 $\times 10^{- 6}$	23	92	0.07296	5.22 $\times 10^{- 6}$	14	56	0.09499	8.69 $\times 10^{- 6}$
	$x_{4}$	13	55	0.055134	3.9 $\times 10^{- 6}$	23	92	0.075265	8.99 $\times 10^{- 6}$	15	60	0.07696	8.5 $\times 10^{- 6}$
	$x_{5}$	12	51	0.047551	8.02 $\times 10^{- 6}$	23	92	0.073937	8.93 $\times 10^{- 6}$	33	133	0.18625	6.45 $\times 10^{- 6}$
	$x_{6}$	13	55	0.055069	9.95 $\times 10^{- 6}$	23	92	0.099888	6.64 $\times 10^{- 6}$	33	133	0.15548	7.51 $\times 10^{- 6}$
50,000	$x_{1}$	12	51	0.19938	5.47 $\times 10^{- 6}$	21	84	0.27031	9.97 $\times 10^{- 6}$	14	56	0.23642	3.51 $\times 10^{- 6}$
	$x_{2}$	13	55	0.22499	3.02 $\times 10^{- 6}$	22	88	0.2657	9.9 $\times 10^{- 6}$	14	56	0.24813	7.12 $\times 10^{- 6}$
	$x_{3}$	13	55	0.19396	6.89 $\times 10^{- 6}$	24	96	0.3246	5.85 $\times 10^{- 6}$	15	60	0.27049	6.53 $\times 10^{- 6}$
	$x_{4}$	13	55	0.20259	8.72 $\times 10^{- 6}$	25	100	0.32373	5.04 $\times 10^{- 6}$	34	137	0.54545	7.13 $\times 10^{- 6}$
	$x_{5}$	13	55	0.19452	5.01 $\times 10^{- 6}$	25	100	0.33764	5.01 $\times 10^{- 6}$	68	274	1.02330	9.99 $\times 10^{- 6}$
	$x_{6}$	14	59	0.22015	6.22 $\times 10^{- 6}$	24	96	0.33687	7.44 $\times 10^{- 6}$	69	278	1.03810	8.05 $\times 10^{- 6}$
100,000	$x_{1}$	12	51	0.39983	7.74 $\times 10^{- 6}$	22	88	0.63809	7.06 $\times 10^{- 6}$	14	56	0.45475	4.96 $\times 10^{- 6}$
	$x_{2}$	13	55	0.32765	4.28 $\times 10^{- 6}$	23	92	0.63458	7.02 $\times 10^{- 6}$	15	60	0.49018	3.39 $\times 10^{- 6}$
	$x_{3}$	13	55	0.30133	9.75 $\times 10^{- 6}$	24	96	0.71422	8.27 $\times 10^{- 6}$	15	60	0.49016	9.24 $\times 10^{- 6}$
	$x_{4}$	14	59	0.42865	3.45 $\times 10^{- 6}$	25	100	0.73524	7.13 $\times 10^{- 6}$	139	559	4.03110	9.01 $\times 10^{- 6}$
	$x_{5}$	13	55	0.34512	7.09 $\times 10^{- 6}$	25	100	0.70625	7.09 $\times 10^{- 6}$	70	282	2.07100	8.54 $\times 10^{- 6}$
	$x_{6}$	14	59	0.40387	8.8 $\times 10^{- 6}$	25	100	0.76777	5.27 $\times 10^{- 6}$	139	559	4.02440	9.38 $\times 10^{- 6}$

Table 4. Numerical Results for Algorithm 1 (DCG), PCG and PDY for Problem 4 with given initial points and dimensions.

		Algorithm 1				PCG				PDY
DIMENSION	INITIAL POINT	ITER	FVAL	TIME	NORM	ITER	FVAL	TIME	NORM	ITER	FVAL	TIME	NORM
1000	$x_{1}$	10	43	0.15461	8.33 $\times 10^{- 6}$	18	72	0.11853	9.93 $\times 10^{- 6}$	12	48	0.00989	4.60 $\times 10^{- 6}$
	$x_{2}$	11	47	0.006276	3.84 $\times 10^{- 6}$	19	76	0.014318	8.75 $\times 10^{- 6}$	12	48	0.00966	9.57 $\times 10^{- 6}$
	$x_{3}$	11	47	0.009859	3.91 $\times 10^{- 6}$	20	80	0.0093776	7.15 $\times 10^{- 6}$	13	52	0.00887	8.49 $\times 10^{- 6}$
	$x_{4}$	11	47	0.007976	5.21 $\times 10^{- 6}$	47	189	0.023321	7.83 $\times 10^{- 6}$	12	48	0.01207	5.83 $\times 10^{- 6}$
	$x_{5}$	12	51	0.008382	4.09 $\times 10^{- 6}$	46	185	0.047105	9.76 $\times 10^{- 6}$	29	117	0.05371	9.43 $\times 10^{- 6}$
	$x_{6}$	12	51	0.008645	3.32 $\times 10^{- 6}$	41	165	0.027719	8.77 $\times 10^{- 6}$	29	117	0.02396	6.65 $\times 10^{- 6}$
5000	$x_{1}$	11	47	0.022024	5.21 $\times 10^{- 6}$	20	80	0.029445	5.57 $\times 10^{- 6}$	13	52	0.02503	3.49 $\times 10^{- 6}$
	$x_{2}$	11	47	0.020587	8.59 $\times 10^{- 6}$	20	80	0.033115	9.8 $\times 10^{- 6}$	13	52	0.02626	7.24 $\times 10^{- 6}$
	$x_{3}$	11	47	0.023714	8.75 $\times 10^{- 6}$	21	84	0.033318	8.01 $\times 10^{- 6}$	14	56	0.03349	6.29 $\times 10^{- 6}$
	$x_{4}$	12	51	0.024728	3.26 $\times 10^{- 6}$	49	197	0.071715	9.46 $\times 10^{- 6}$	13	52	0.02258	4.25 $\times 10^{- 6}$
	$x_{5}$	12	51	0.031015	9.14 $\times 10^{- 6}$	49	197	0.068565	8.68 $\times 10^{- 6}$	31	125	0.05471	7.59 $\times 10^{- 6}$
	$x_{6}$	12	51	0.030012	7.43 $\times 10^{- 6}$	44	177	0.070862	7.79 $\times 10^{- 6}$	63	254	0.10064	8.54 $\times 10^{- 6}$
10,000	$x_{1}$	11	47	0.041476	7.37 $\times 10^{- 6}$	20	80	0.043013	7.88 $\times 10^{- 6}$	13	52	0.03761	4.93 $\times 10^{- 6}$
	$x_{2}$	12	51	0.047866	3.4 $\times 10^{- 6}$	21	84	0.051685	6.94 $\times 10^{- 6}$	14	56	0.04100	3.37 $\times 10^{- 6}$
	$x_{3}$	12	51	0.042607	3.46 $\times 10^{- 6}$	22	88	0.050422	5.67 $\times 10^{- 6}$	14	56	0.03919	8.90 $\times 10^{- 6}$
	$x_{4}$	12	51	0.036406	4.61 $\times 10^{- 6}$	50	201	0.17563	9.84 $\times 10^{- 6}$	32	129	0.09613	6.02 $\times 10^{- 6}$
	$x_{5}$	13	55	0.041374	3.61 $\times 10^{- 6}$	50	201	0.20035	9.03 $\times 10^{- 6}$	32	129	0.09177	6.44 $\times 10^{- 6}$
	$x_{6}$	13	55	0.039847	2.94 $\times 10^{- 6}$	45	181	0.12214	8.11 $\times 10^{- 6}$	64	258	0.20791	9.39 $\times 10^{- 6}$
50,000	$x_{1}$	12	51	0.13928	4.61 $\times 10^{- 6}$	21	84	0.27145	8.83 $\times 10^{- 6}$	14	56	0.17193	3.63 $\times 10^{- 6}$
	$x_{2}$	12	51	0.18031	7.6 $\times 10^{- 6}$	22	88	0.23149	7.78 $\times 10^{- 6}$	14	56	0.15237	7.54 $\times 10^{- 6}$
	$x_{3}$	12	51	0.12526	7.74 $\times 10^{- 6}$	23	92	0.28789	6.36 $\times 10^{- 6}$	15	60	0.16549	6.66 $\times 10^{- 6}$
	$x_{4}$	13	55	0.14322	2.88 $\times 10^{- 6}$	53	213	0.61624	8.75 $\times 10^{- 6}$	67	270	0.76283	7.81 $\times 10^{- 6}$
	$x_{5}$	13	55	0.17904	8.08 $\times 10^{- 6}$	53	213	0.7119	8.02 $\times 10^{- 6}$	67	270	0.76157	8.80 $\times 10^{- 6}$
	$x_{6}$	13	55	0.13635	6.57 $\times 10^{- 6}$	47	189	0.48192	9.8 $\times 10^{- 6}$	269	1080	2.92510	9.41 $\times 10^{- 6}$
100,000	$x_{1}$	12	51	0.24293	6.52 $\times 10^{- 6}$	22	88	0.60822	6.25 $\times 10^{- 6}$	14	56	0.30229	5.13 $\times 10^{- 6}$
	$x_{2}$	13	55	0.27433	3.01 $\times 10^{- 6}$	23	92	0.52965	5.51 $\times 10^{- 6}$	15	60	0.31648	3.59 $\times 10^{- 6}$
	$x_{3}$	13	55	0.2714	3.06 $\times 10^{- 6}$	23	92	0.57064	8.99 $\times 10^{- 6}$	32	129	0.72838	9.99 $\times 10^{- 6}$
	$x_{4}$	13	55	0.26819	4.08 $\times 10^{- 6}$	54	217	1.1805	9.1 $\times 10^{- 6}$	135	543	2.86780	9.73 $\times 10^{- 6}$
	$x_{5}$	14	59	0.31696	3.2 $\times 10^{- 6}$	54	217	1.107	8.34 $\times 10^{- 6}$	272	1092	5.74140	9.91 $\times 10^{- 6}$
	$x_{6}$	13	55	0.2698	9.29 $\times 10^{- 6}$	49	197	1.0617	7.49 $\times 10^{- 6}$	548	2197	11.44130	9.87 $\times 10^{- 6}$

Table 5. Numerical Results for Algorithm 1 (DCG), PCG and PDY for Problem 5 with given initial points and dimensions.

		Algorithm 1				PCG				PDY
DIMENSION	INITIAL POINT	ITER	FVAL	TIME	NORM	ITER	FVAL	TIME	NORM	ITER	FVAL	TIME	NORM
1000	$x_{1}$	19	78	0.71709	8.63 $\times 10^{- 6}$	22	83	0.099338	7.48 $\times 10^{- 6}$	16	63	0.07575	6.03 $\times 10^{- 6}$
	$x_{2}$	21	86	0.017127	7.65 $\times 10^{- 6}$	23	88	0.016014	7.31 $\times 10^{- 6}$	16	63	0.01470	5.42 $\times 10^{- 6}$
	$x_{3}$	23	95	0.013909	7.23 $\times 10^{- 6}$	23	90	0.016328	9.31 $\times 10^{- 6}$	33	132	0.02208	6.75 $\times 10^{- 6}$
	$x_{4}$	22	92	0.0165	8.64 $\times 10^{- 6}$	49	197	0.030124	8.45 $\times 10^{- 6}$	30	121	0.01835	8.39 $\times 10^{- 6}$
	$x_{5}$	35	145	0.024702	8.26 $\times 10^{- 6}$	53	213	0.039321	8.38 $\times 10^{- 6}$	32	129	0.02700	8.47 $\times 10^{- 6}$
	$x_{6}$	43	182	0.027471	8.7 $\times 10^{- 6}$	46	185	0.033627	8.8 $\times 10^{- 6}$	30	121	0.01712	6.95 $\times 10^{- 6}$
5000	$x_{1}$	146	592	0.23803	9.45 $\times 10^{- 6}$	24	91	0.060158	6.36 $\times 10^{- 6}$	17	67	0.04394	5.64 $\times 10^{- 6}$
	$x_{2}$	21	86	0.04337	9.46 $\times 10^{- 6}$	25	95	0.060385	6.24 $\times 10^{- 6}$	17	67	0.04635	5.07 $\times 10^{- 6}$
	$x_{3}$	24	99	0.054619	8.27 $\times 10^{- 6}$	25	98	0.040015	5.86 $\times 10^{- 6}$	35	140	0.08311	9.74 $\times 10^{- 6}$
	$x_{4}$	24	100	0.066424	6.66 $\times 10^{- 6}$	53	213	0.098097	9.11 $\times 10^{- 6}$	33	133	0.08075	6.02 $\times 10^{- 6}$
	$x_{5}$	38	157	0.071222	9.28 $\times 10^{- 6}$	58	233	0.10958	8.56 $\times 10^{- 6}$	35	141	0.10091	7.51 $\times 10^{- 6}$
	$x_{6}$	45	190	0.090276	7.14 $\times 10^{- 6}$	50	201	0.21521	7.65 $\times 10^{- 6}$	32	129	0.08054	8.55 $\times 10^{- 6}$
10,000	$x_{1}$	211	853	0.60357	9.65 $\times 10^{- 6}$	25	95	0.076427	5.4 $\times 10^{- 6}$	17	67	0.06816	8.81 $\times 10^{- 6}$
	$x_{2}$	22	90	0.08012	4.98 $\times 10^{- 6}$	25	95	0.098461	8.9 $\times 10^{- 6}$	17	67	0.08833	7.80 $\times 10^{- 6}$
	$x_{3}$	25	103	0.089269	5.89 $\times 10^{- 6}$	25	98	0.07495	8.64 $\times 10^{- 6}$	37	148	0.14732	6.36 $\times 10^{- 6}$
	$x_{4}$	25	104	0.11781	5.54 $\times 10^{- 6}$	55	221	0.19048	9.11 $\times 10^{- 6}$	37	149	0.14293	8.25 $\times 10^{- 6}$
	$x_{5}$	40	165	0.15859	7.43 $\times 10^{- 6}$	60	241	0.19751	9.01 $\times 10^{- 6}$	36	145	0.14719	8.23 $\times 10^{- 6}$
	$x_{6}$	46	194	0.1728	8.62 $\times 10^{- 6}$	51	205	0.28882	9.62 $\times 10^{- 6}$	74	298	0.26456	7.79 $\times 10^{- 6}$
50,000	$x_{1}$	225	909	2.1373	9.93 $\times 10^{- 6}$	26	99	0.34575	6.75 $\times 10^{- 6}$	42	169	0.58113	7.78 $\times 10^{- 6}$
	$x_{2}$	23	94	0.31098	4.48 $\times 10^{- 6}$	27	103	0.43806	5.16 $\times 10^{- 6}$	42	169	0.58456	7.13 $\times 10^{- 6}$
	$x_{3}$	26	107	0.36293	6.83 $\times 10^{- 6}$	27	106	0.4815	5.28 $\times 10^{- 6}$	41	165	0.58717	8.87 $\times 10^{- 6}$
	$x_{4}$	26	108	0.32427	9.72 $\times 10^{- 6}$	60	241	0.90868	8.66 $\times 10^{- 6}$	40	161	0.56431	7.17 $\times 10^{- 6}$
	$x_{5}$	43	177	0.48938	9.47 $\times 10^{- 6}$	65	261	0.7924	9.05 $\times 10^{- 6}$	82	330	1.08920	8.44 $\times 10^{- 6}$
	$x_{6}$	50	210	0.69117	8.12 $\times 10^{- 6}$	56	225	0.72334	8.19 $\times 10^{- 6}$	80	322	1.06670	7.82 $\times 10^{- 6}$
100,000	$x_{1}$	231	933	4.2588	9.85 $\times 10^{- 6}$	26	99	0.71242	9.73 $\times 10^{- 6}$	43	173	1.09620	8.47 $\times 10^{- 6}$
	$x_{2}$	139	564	2.7266	9.96 $\times 10^{- 6}$	27	103	0.62746	7.39 $\times 10^{- 6}$	43	173	1.10040	7.77 $\times 10^{- 6}$
	$x_{3}$	26	107	0.57505	9.92 $\times 10^{- 6}$	27	106	0.82989	7.77 $\times 10^{- 6}$	42	169	1.08330	9.66 $\times 10^{- 6}$
	$x_{4}$	27	112	0.62227	8.52 $\times 10^{- 6}$	62	249	1.5474	9 $\times 10^{- 6}$	85	342	2.11880	9.22 $\times 10^{- 6}$
	$x_{5}$	45	185	0.8992	7.79 $\times 10^{- 6}$	67	269	1.6692	9.5 $\times 10^{- 6}$	84	338	2.10640	9.78 $\times 10^{- 6}$
	$x_{6}$	52	218	1.4318	7.37 $\times 10^{- 6}$	58	233	1.4333	8.32 $\times 10^{- 6}$	167	671	4.06200	9.90 $\times 10^{- 6}$

Table 6. Numerical Results for Algorithm 1 (DCG), PCG and PDY for Problem 6 with given initial points and dimensions.

		Algorithm 1				PCG				PDY
DIMENSION	INITIAL POINT	ITER	FVAL	TIME	NORM	ITER	FVAL	TIME	NORM	ITER	FVAL	TIME	NORM
1000	$x_{1}$	13	55	1.38	5.68 $\times 10^{- 6}$	23	92	0.4038	9.28 $\times 10^{- 6}$	15	60	0.01671	4.35 $\times 10^{- 6}$
	$x_{2}$	13	55	0.013339	5.47 $\times 10^{- 6}$	23	92	0.016325	8.92 $\times 10^{- 6}$	15	60	0.01346	4.18 $\times 10^{- 6}$
	$x_{3}$	13	55	0.066142	4.81 $\times 10^{- 6}$	23	92	0.023045	7.86 $\times 10^{- 6}$	15	60	0.01630	3.68 $\times 10^{- 6}$
	$x_{4}$	13	55	0.026838	3.3 $\times 10^{- 6}$	23	92	0.016172	5.38 $\times 10^{- 6}$	14	56	0.01339	7.48 $\times 10^{- 6}$
	$x_{5}$	12	51	0.009864	9.45 $\times 10^{- 6}$	22	88	0.03785	8.62 $\times 10^{- 6}$	14	56	0.01267	6.01 $\times 10^{- 6}$
	$x_{1}$	12	51	0.009881	5.57 $\times 10^{- 6}$	22	88	0.015013	5.08 $\times 10^{- 6}$	14	56	0.01685	3.54 $\times 10^{- 6}$
5000	$x_{1}$	14	59	0.042533	3.56 $\times 10^{- 6}$	25	100	0.061642	5.22 $\times 10^{- 6}$	15	60	0.05038	9.73 $\times 10^{- 6}$
	$x_{2}$	14	59	0.036648	3.43 $\times 10^{- 6}$	25	100	0.092952	5.02 $\times 10^{- 6}$	15	60	0.04775	9.36 $\times 10^{- 6}$
	$x_{3}$	14	59	0.043452	3.02 $\times 10^{- 6}$	24	96	0.068141	8.82 $\times 10^{- 6}$	15	60	0.04923	8.25 $\times 10^{- 6}$
	$x_{4}$	13	55	0.032579	7.38 $\times 10^{- 6}$	24	96	0.084625	6.04 $\times 10^{- 6}$	15	60	0.05793	5.64 $\times 10^{- 6}$
	$x_{5}$	13	55	0.03295	5.92 $\times 10^{- 6}$	23	92	0.086122	9.67 $\times 10^{- 6}$	15	60	0.04597	4.53 $\times 10^{- 6}$
	$x_{6}$	13	55	0.033062	3.49 $\times 10^{- 6}$	23	92	0.093318	5.7 $\times 10^{- 6}$	14	56	0.05070	7.93 $\times 10^{- 6}$
10,000	$x_{1}$	14	59	0.064917	5.04 $\times 10^{- 6}$	25	100	0.21424	7.38 $\times 10^{- 6}$	68	274	0.40724	9.06 $\times 10^{- 6}$
	$x_{2}$	14	59	0.069913	4.84 $\times 10^{- 6}$	25	100	0.13978	7.09 $\times 10^{- 6}$	68	274	0.41818	8.72 $\times 10^{- 6}$
	$x_{3}$	14	59	0.08473	4.27 $\times 10^{- 6}$	25	100	0.1731	6.25 $\times 10^{- 6}$	34	137	0.21905	6.22 $\times 10^{- 6}$
	$x_{4}$	14	59	0.075847	2.92 $\times 10^{- 6}$	24	96	0.14744	8.54 $\times 10^{- 6}$	15	60	0.10076	7.98 $\times 10^{- 6}$
	$x_{5}$	13	55	0.07974	8.38 $\times 10^{- 6}$	24	96	0.14169	6.85 $\times 10^{- 6}$	15	60	0.12680	6.40 $\times 10^{- 6}$
	$x_{6}$	13	55	0.063129	4.94 $\times 10^{- 6}$	23	92	0.15294	8.06 $\times 10^{- 6}$	15	60	0.11984	3.78 $\times 10^{- 6}$
50,000	$x_{1}$	15	63	0.25329	3.15 $\times 10^{- 6}$	26	104	0.64669	8.26 $\times 10^{- 6}$	143	575	3.09120	9.42 $\times 10^{- 6}$
	$x_{2}$	15	63	0.36394	3.03 $\times 10^{- 6}$	26	104	0.67717	7.95 $\times 10^{- 6}$	143	575	3.06200	9.06 $\times 10^{- 6}$
	$x_{3}$	14	59	0.2413	9.54 $\times 10^{- 6}$	26	104	0.5562	7 $\times 10^{- 6}$	142	571	3.04950	9.04 $\times 10^{- 6}$
	$x_{4}$	14	59	0.27502	6.53 $\times 10^{- 6}$	25	100	0.56171	9.56 $\times 10^{- 6}$	69	278	1.53920	9.14 $\times 10^{- 6}$
	$x_{5}$	14	59	0.36404	5.24 $\times 10^{- 6}$	25	100	0.57982	7.67 $\times 10^{- 6}$	68	274	1.49490	9.43 $\times 10^{- 6}$
	$x_{6}$	14	59	0.2506	3.09 $\times 10^{- 6}$	24	96	0.58645	9.03 $\times 10^{- 6}$	15	60	0.38177	8.44 $\times 10^{- 6}$
100,000	$x_{1}$	15	63	0.84781	4.45 $\times 10^{- 6}$	27	108	1.3215	5.86 $\times 10^{- 6}$	292	1172	13.59530	9.53 $\times 10^{- 6}$
	$x_{2}$	15	63	0.66663	4.28 $\times 10^{- 6}$	27	108	1.5062	5.63 $\times 10^{- 6}$	290	1164	13.30930	9.75 $\times 10^{- 6}$
	$x_{3}$	15	63	0.66683	3.77 $\times 10^{- 6}$	26	104	1.166	9.9 $\times 10^{- 6}$	144	579	6.68150	9.96 $\times 10^{- 6}$
	$x_{4}$	14	59	0.62697	9.24 $\times 10^{- 6}$	26	104	1.3961	6.78 $\times 10^{- 6}$	141	567	6.50800	9.92 $\times 10^{- 6}$
	$x_{5}$	14	59	0.62891	7.41 $\times 10^{- 6}$	26	104	1.2711	5.44 $\times 10^{- 6}$	70	282	3.30510	8.07 $\times 10^{- 6}$
	$x_{6}$	14	59	0.62422	4.37 $\times 10^{- 6}$	25	100	1.1685	6.4 $\times 10^{- 6}$	34	137	1.64510	6.37 $\times 10^{- 6}$

Table 7. Numerical Results for Algorithm 1 (DCG), PCG and PDY for Problem 7 with given initial points and dimensions.

		Algorithm 1				PCG				PDY
DIMENSION	INITIAL POINT	ITER	FVAL	TIME	NORM	ITER	FVAL	TIME	NORM	ITER	FVAL	TIME	NORM
1000	$x_{1}$	6	28	0.25689	2 $\times 10^{- 6}$	17	69	1.2275	6.98 $\times 10^{- 6}$	14	57	0.00953	5.28 $\times 10^{- 6}$
	$x_{2}$	6	28	0.008469	1.26 $\times 10^{- 6}$	15	61	0.23396	9.89 $\times 10^{- 6}$	13	53	0.00896	9.05 $\times 10^{- 6}$
	$x_{3}$	4	20	0.003619	9.25 $\times 10^{- 6}$	16	65	0.008095	5.79 $\times 10^{- 6}$	3	12	0.00426	8.47 $\times 10^{- 6}$
	$x_{4}$	5	24	0.004345	5.7 $\times 10^{- 6}$	16	65	0.010077	5.21 $\times 10^{- 6}$	15	61	0.01169	6.73 $\times 10^{- 6}$
	$x_{5}$	6	28	0.007146	4.42 $\times 10^{- 6}$	19	77	0.05354	4.95 $\times 10^{- 6}$	31	126	0.03646	9.03 $\times 10^{- 6}$
	$x_{6}$	6	27	0.004299	4.43 $\times 10^{- 6}$	18	72	0.025677	8.93 $\times 10^{- 6}$	15	60	0.01082	3.99 $\times 10^{- 6}$
5000	$x_{1}$	6	28	0.012915	4.47 $\times 10^{- 6}$	18	73	0.17722	7.6 $\times 10^{- 6}$	15	61	0.03215	4.25 $\times 10^{- 6}$
	$x_{2}$	6	28	0.012272	2.81 $\times 10^{- 6}$	17	69	0.027729	5.25 $\times 10^{- 6}$	14	57	0.02942	7.40 $\times 10^{- 6}$
	$x_{3}$	5	24	0.014669	1.16 $\times 10^{- 6}$	17	69	0.02985	6.31 $\times 10^{- 6}$	4	16	0.01107	1.01 $\times 10^{- 7}$
	$x_{4}$	6	28	0.012765	7.14 $\times 10^{- 7}$	17	69	0.028176	5.68 $\times 10^{- 6}$	16	65	0.04331	5.43 $\times 10^{- 6}$
	$x_{5}$	6	28	0.01331	9.89 $\times 10^{- 6}$	20	81	0.032213	5.39 $\times 10^{- 6}$	33	134	0.09379	7.78 $\times 10^{- 6}$
	$x_{6}$	6	27	0.015828	9.91 $\times 10^{- 6}$	19	76	0.044328	9.73 $\times 10^{- 6}$	15	60	0.04077	8.92 $\times 10^{- 6}$
10,000	$x_{1}$	6	28	0.022346	6.32 $\times 10^{- 6}$	19	77	0.17863	5.23 $\times 10^{- 6}$	15	61	0.06484	6.01 $\times 10^{- 6}$
	$x_{2}$	6	28	0.022669	3.97 $\times 10^{- 6}$	17	69	0.049242	7.42 $\times 10^{- 6}$	15	61	0.07734	3.77 $\times 10^{- 6}$
	$x_{3}$	5	24	0.039342	1.64 $\times 10^{- 6}$	17	69	0.048238	8.92 $\times 10^{- 6}$	4	16	0.02707	1.42 $\times 10^{- 7}$
	$x_{4}$	6	28	0.021017	1.01 $\times 10^{- 6}$	17	69	0.04807	8.03 $\times 10^{- 6}$	16	65	0.07941	7.69 $\times 10^{- 6}$
	$x_{5}$	7	32	0.031654	7.83 $\times 10^{- 7}$	20	81	0.063156	7.62 $\times 10^{- 6}$	34	138	0.14942	6.83 $\times 10^{- 6}$
	$x_{6}$	7	31	0.023456	7.85 $\times 10^{- 7}$	20	80	0.059438	6.7 $\times 10^{- 6}$	34	138	0.15224	8.81 $\times 10^{- 6}$
50,000	$x_{1}$	7	32	0.092452	7.91 $\times 10^{- 7}$	20	81	1.0808	5.7 $\times 10^{- 6}$	16	65	0.25995	4.89 $\times 10^{- 6}$
	$x_{2}$	6	28	0.1068	8.88 $\times 10^{- 6}$	18	73	0.32804	8.08 $\times 10^{- 6}$	15	61	0.24674	8.42 $\times 10^{- 6}$
	$x_{3}$	5	24	0.065684	3.66 $\times 10^{- 6}$	18	73	0.2189	9.71 $\times 10^{- 6}$	4	16	0.09405	3.18 $\times 10^{- 7}$
	$x_{4}$	6	28	0.10193	2.26 $\times 10^{- 6}$	18	73	0.3497	8.75 $\times 10^{- 6}$	36	146	0.55207	6.39 $\times 10^{- 6}$
	$x_{5}$	7	32	0.095676	1.75 $\times 10^{- 6}$	21	85	0.22595	8.3 $\times 10^{- 6}$	35	142	0.54679	9.05 $\times 10^{- 6}$
	$x_{6}$	7	31	0.092855	1.76 $\times 10^{- 6}$	21	84	0.22374	7.3 $\times 10^{- 6}$	36	146	0.55764	7.59 $\times 10^{- 6}$
100,000	$x_{1}$	7	32	0.17597	1.12 $\times 10^{- 6}$	20	81	2.1675	8.06 $\times 10^{- 6}$	17	69	0.52595	5.68 $\times 10^{- 6}$
	$x_{2}$	7	32	0.1741	7.03 $\times 10^{- 7}$	19	77	0.45553	5.57 $\times 10^{- 6}$	16	65	0.52102	4.34 $\times 10^{- 6}$
	$x_{3}$	5	24	0.17522	5.18 $\times 10^{- 6}$	19	77	0.43219	6.69 $\times 10^{- 6}$	4	16	0.14864	4.50 $\times 10^{- 7}$
	$x_{4}$	6	28	0.20785	3.19 $\times 10^{- 6}$	19	77	0.52259	6.03 $\times 10^{- 6}$	36	146	1.05360	9.04 $\times 10^{- 6}$
	$x_{5}$	7	32	0.23979	2.48 $\times 10^{- 6}$	22	89	0.6171	5.72 $\times 10^{- 6}$	74	299	2.10730	8.55 $\times 10^{- 6}$
	$x_{6}$	7	31	0.23128	2.48 $\times 10^{- 6}$	22	88	0.57384	5.03 $\times 10^{- 6}$	37	150	1.08240	6.66 $\times 10^{- 6}$

Table 8. Numerical Results for Algorithm 1 (DCG), PCG and PDY for Problem 8 with given initial points and dimensions.

		Algorithm 1				PCG				PDY
DIMENSION	INITIAL POINT	ITER	FVAL	TIME	NORM	ITER	FVAL	TIME	NORM	ITER	FVAL	TIME	NORM
1000	$x_{1}$	7	28	0.11495	3.03 $\times 10^{- 6}$	9	32	0.85797	7.6 $\times 10^{- 6}$	69	279	0.05538	8.95 $\times 10^{- 6}$
	$x_{2}$	7	28	0.005034	3.03 $\times 10^{- 6}$	9	32	0.034675	7.6 $\times 10^{- 6}$	270	1085	0.18798	9.72 $\times 10^{- 6}$
	$x_{3}$	7	28	0.006743	3.03 $\times 10^{- 6}$	9	32	0.005985	7.6 $\times 10^{- 6}$	24	52	0.02439	6.57 $\times 10^{- 6}$
	$x_{4}$	7	28	0.005856	3.03 $\times 10^{- 6}$	9	32	0.004808	7.6 $\times 10^{- 6}$	27	58	0.01520	7.59 $\times 10^{- 6}$
	$x_{5}$	7	28	0.004635	3.03 $\times 10^{- 6}$	9	32	0.015026	7.6 $\times 10^{- 6}$	28	61	0.04330	9.21 $\times 10^{- 6}$
	$x_{6}$	7	28	0.006487	3.03 $\times 10^{- 6}$	9	32	0.15778	7.6 $\times 10^{- 6}$	40	85	0.02116	8.45 $\times 10^{- 6}$
5000	$x_{1}$	5	22	0.009068	4.52 $\times 10^{- 6}$	7	26	0.67239	1.3 $\times 10^{- 6}$	658	2639	1.13030	9.98 $\times 10^{- 6}$
	$x_{2}$	5	22	0.009369	4.52 $\times 10^{- 6}$	7	26	0.010651	1.3 $\times 10^{- 6}$	27	58	0.05101	7.59 $\times 10^{- 6}$
	$x_{3}$	5	22	0.010895	4.52 $\times 10^{- 6}$	7	26	0.015758	1.3 $\times 10^{- 6}$	49	104	0.08035	8.11 $\times 10^{- 6}$
	$x_{4}$	5	22	0.014958	4.52 $\times 10^{- 6}$	7	26	0.014935	1.3 $\times 10^{- 6}$	40	85	0.07979	8.45 $\times 10^{- 6}$
	$x_{5}$	5	22	0.01507	4.52 $\times 10^{- 6}$	7	26	0.01524	1.3 $\times 10^{- 6}$	18	40	0.09128	9.14 $\times 10^{- 6}$
	$x_{6}$	5	22	0.008716	4.52 $\times 10^{- 6}$	7	26	0.1999	1.3 $\times 10^{- 6}$	17	38	0.18528	8.98 $\times 10^{- 6}$
10,000	$x_{1}$	6	27	0.031198	3.81 $\times 10^{- 6}$	5	19	0.044387	5.06 $\times 10^{- 6}$	49	104	0.20443	7.62 $\times 10^{- 6}$
	$x_{2}$	6	27	0.02098	3.81 $\times 10^{- 6}$	5	19	0.0223	5.06 $\times 10^{- 6}$	40	85	0.15801	8.45 $\times 10^{- 6}$
	$x_{3}$	6	27	0.01991	3.81 $\times 10^{- 6}$	5	19	0.018209	5.06 $\times 10^{- 6}$	19	42	0.37880	7.66 $\times 10^{- 6}$
	$x_{4}$	6	27	0.025402	3.81 $\times 10^{- 6}$	5	19	0.021654	5.06 $\times 10^{- 6}$	90	187	1.25802	9.7 $\times 10^{- 6}$
	$x_{5}$	6	27	0.025816	3.81 $\times 10^{- 6}$	5	19	0.017353	5.06 $\times 10^{- 6}$	988	1988	12.68259	9.93 $\times 10^{- 6}$
	$x_{6}$	6	27	0.025065	3.81 $\times 10^{- 6}$	5	19	0.019763	5.06 $\times 10^{- 6}$	27	58	0.32859	7.59 $\times 10^{- 6}$
50,000	$x_{1}$	4	21	0.083641	2.34 $\times 10^{- 7}$	8	33	0.42902	5.15 $\times 10^{- 6}$	19	42	0.52291	6.42 $\times 10^{- 6}$
	$x_{2}$	4	21	0.074156	2.34 $\times 10^{- 7}$	8	33	0.11525	5.15 $\times 10^{- 6}$	148	304	3.93063	9.92 $\times 10^{- 6}$
	$x_{3}$	4	21	0.078596	2.34 $\times 10^{- 7}$	8	33	0.14432	5.15 $\times 10^{- 6}$	937	1886	22.97097	9.87 $\times 10^{- 6}$
	$x_{4}$	4	21	0.078289	2.34 $\times 10^{- 7}$	8	33	0.11562	5.15 $\times 10^{- 6}$	27	58	0.68467	7.59 $\times 10^{- 6}$
	$x_{5}$	4	21	0.073535	2.34 $\times 10^{- 7}$	8	33	0.11674	5.15 $\times 10^{- 6}$	346	702	8.45043	9.79 $\times 10^{- 6}$
	$x_{6}$	4	21	0.081909	2.34 $\times 10^{- 7}$	8	33	0.10486	5.15 $\times 10^{- 6}$	40	85	0.99230	8.45 $\times 10^{- 6}$
100,000	$x_{1}$	4	22	0.1663	6.25 $\times 10^{- 6}$	6	25	1.2922	6.81 $\times 10^{- 7}$	-	-	-	-
	$x_{2}$	4	22	0.15147	6.25 $\times 10^{- 6}$	6	25	0.18839	6.81 $\times 10^{- 7}$	-	-	-	-
	$x_{3}$	4	22	0.15582	6.25 $\times 10^{- 6}$	6	25	0.16153	6.81 $\times 10^{- 7}$	-	-	-	-
	$x_{4}$	4	22	0.15465	6.25 $\times 10^{- 6}$	6	25	0.17397	6.81 $\times 10^{- 7}$	-	-	-	-
	$x_{5}$	4	22	0.16744	6.25 $\times 10^{- 6}$	6	25	0.18586	6.81 $\times 10^{- 7}$	-	-	-	-
	$x_{6}$	4	22	0.1687	6.25 $\times 10^{- 6}$	6	25	0.17938	6.81 $\times 10^{- 7}$	-	-	-	-

Table 9. Numerical Results for Algorithm 1 (DCG), PCG and PDY for Problem 9 with given initial points and dimensions.

		Algorithm 1				PCG				PDY
DIMENSION	INITIAL POINT	ITER	FVAL	TIME	NORM	ITER	FVAL	TIME	NORM	ITER	FVAL	TIME	NORM
4	$x_{1}$	51	215	0.23665	9.01 $\times 10^{- 6}$	79	321	0.5978	9.76 $\times 10^{- 6}$	59	241	0.71268	9.36 $\times 10^{- 6}$
	$x_{2}$	51	215	0.04968	9.99 $\times 10^{- 6}$	77	313	0.016326	9.85 $\times 10^{- 6}$	58	237	0.045441	9.73 $\times 10^{- 6}$
	$x_{3}$	53	223	0.017211	9.46 $\times 10^{- 6}$	80	325	0.16529	9.38 $\times 10^{- 6}$	59	241	0.019552	9.9 $\times 10^{- 6}$
	$x_{4}$	53	223	0.019004	9.68 $\times 10^{- 6}$	83	337	0.041713	9.57 $\times 10^{- 6}$	62	253	0.022007	8.07 $\times 10^{- 6}$
	$x_{5}$	57	239	0.023447	8.87 $\times 10^{- 6}$	81	329	0.11972	9.04 $\times 10^{- 6}$	61	249	0.040117	8.36 $\times 10^{- 6}$
	$x_{6}$	54	227	0.020832	9.31 $\times 10^{- 6}$	82	333	0.016127	9.3 $\times 10^{- 6}$	61	249	0.017374	9.18 $\times 10^{- 6}$

Table 10. Efficiency comparison based on the value of the number of iterations (Iter), objective function (ObjFun) value, mean-square-error (MSE) and signal-to-noise-ratio (SNR) under different Pi (

σ

σ

Image	Iter		ObjFun		MSE		SNR
	DCG	CGD	DCG	CGD	DCG	CGD	DCG	CGD
P1(1E-8)	8	9	4.397 $\times 10^{3}$	4.398 $\times 10^{3}$	3.136 $\times 10^{- 2}$	3.157 $\times 10^{- 2}$	9.42	9.39
P1(1E-1)	8	9	4.399 $\times 10^{3}$	4.401 $\times 10^{3}$	3.147 $\times 10^{- 2}$	3.163 $\times 10^{- 2}$	9.40	9.38
P1(0.11)	11	8	4.428 $\times 10^{3}$	4.432 $\times 10^{3}$	3.229 $\times 10^{- 2}$	3.232 $\times 10^{- 2}$	9.29	9.29
P1(0.25)	12	8	4.468 $\times 10^{3}$	4.473 $\times 10^{3}$	3.365 $\times 10^{- 2}$	3.289 $\times 10^{- 2}$	9.11	9.21
P1(1E-8)	9	9	4.555 $\times 10^{3}$	4.556 $\times 10^{3}$	3.287 $\times 10^{- 2}$	3.3412 $\times 10^{- 2}$	9.14	9.07
P1(1E-1)	9	9	4.558 $\times 10^{3}$	4.559 $\times 10^{3}$	3.298 $\times 10^{- 2}$	3.348 $\times 10^{- 2}$	9.12	9.06
P1(0.11)	12	12	4.588 $\times 10^{3}$	4.591 $\times 10^{3}$	3.416 $\times 10^{- 2}$	3.446 $\times 10^{- 2}$	8.97	8.93
P1(0.25)	7	8	4.628 $\times 10^{3}$	4.630 $\times 10^{3}$	3.621 $\times 10^{- 2}$	3.500 $\times 10^{- 2}$	8.72	8.86
P1(1E-8)	9	9	5.179 $\times 10^{3}$	5.179 $\times 10^{3}$	3.209 $\times 10^{- 2}$	3.3259 $\times 10^{- 2}$	10.03	9.96
P1(1E-1)	9	9	5.182 $\times 10^{3}$	5.182 $\times 10^{3}$	3.231 $\times 10^{- 2}$	3.267 $\times 10^{- 2}$	10.00	9.95
P1(0.11)	7	9	5.209 $\times 10^{3}$	5.209 $\times 10^{3}$	3.436 $\times 10^{- 2}$	3.344 $\times 10^{- 2}$	9.73	9.85
P1(0.25)	10	8	5.250 $\times 10^{3}$	5.254 $\times 10^{3}$	3.557 $\times 10^{- 2}$	3.438 $\times 10^{- 2}$	9.58	9.73
P1(1E-8)	9	9	4.388 $\times 10^{3}$	4.389 $\times 10^{3}$	3.299 $\times 10^{- 2}$	3.335 $\times 10^{- 2}$	9.03	8.99
P1(1E-1)	9	9	4.391 $\times 10^{3}$	4.393 $\times 10^{3}$	3.308 $\times 10^{- 2}$	3.340 $\times 10^{- 2}$	9.02	8.98
P1(0.11)	12	8	4.421 $\times 10^{3}$	4.424 $\times 10^{3}$	3.425 $\times 10^{- 2}$	3.411 $\times 10^{- 2}$	8.87	8.89
P1(0.25)	7	8	4.461 $\times 10^{3}$	4.463 $\times 10^{3}$	3.621 $\times 10^{- 2}$	3.483 $\times 10^{- 2}$	8.63	8.80

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Abubakar, A.B.; Kumam, P.; Mohammad, H.; Awwal, A.M. An Efficient Conjugate Gradient Method for Convex Constrained Monotone Nonlinear Equations with Applications. Mathematics 2019, 7, 767. https://doi.org/10.3390/math7090767

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Abubakar AB, Kumam P, Mohammad H, Awwal AM. An Efficient Conjugate Gradient Method for Convex Constrained Monotone Nonlinear Equations with Applications. Mathematics. 2019; 7(9):767. https://doi.org/10.3390/math7090767

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Abubakar, Auwal Bala, Poom Kumam, Hassan Mohammad, and Aliyu Muhammed Awwal. 2019. "An Efficient Conjugate Gradient Method for Convex Constrained Monotone Nonlinear Equations with Applications" Mathematics 7, no. 9: 767. https://doi.org/10.3390/math7090767

APA Style

Abubakar, A. B., Kumam, P., Mohammad, H., & Awwal, A. M. (2019). An Efficient Conjugate Gradient Method for Convex Constrained Monotone Nonlinear Equations with Applications. Mathematics, 7(9), 767. https://doi.org/10.3390/math7090767

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An Efficient Conjugate Gradient Method for Convex Constrained Monotone Nonlinear Equations with Applications^†

Abstract

1. Introduction

2. Algorithm: Motivation and Convergence Result

3. Numerical Examples

Applications in Compressive Sensing

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An Efficient Conjugate Gradient Method for Convex Constrained Monotone Nonlinear Equations with Applications †

Abstract

1. Introduction

2. Algorithm: Motivation and Convergence Result

3. Numerical Examples

Applications in Compressive Sensing

4. Conclusions

Author Contributions

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Acknowledgments

Conflicts of Interest

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An Efficient Conjugate Gradient Method for Convex Constrained Monotone Nonlinear Equations with Applications^†