# A Two-Step Spectral Gradient Projection Method for System of Nonlinear Monotone Equations and Image Deblurring Problems

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## Abstract

**:**

## 1. Introduction

## 2. Two Step Iterative Scheme and Its Convergence Analysis

**Definition**

**1.**

- (i)
- monotone if$$\langle F\left(x\right)-F\left(y\right),\phantom{\rule{3.33333pt}{0ex}}x-y\rangle \ge 0.$$
- (ii)
- Lipschitzian continuous if there exists $L>0$ such that$$\parallel F\left(x\right)-F\left(y\right)\parallel \le L\parallel x-y\parallel .$$

**Assumption**

**1.**

**Lemma**

**1.**

**Proof**

**of**

**Lemma**

**1.**

**Remark**

**1.**

- (i)
- From Lemma 1, it is not difficult to see that the two search directions ${d}_{k}^{I}$ and ${d}_{k}^{II}$ satisfy the descent condition. That is,$$\left\{\begin{array}{c}\langle {d}_{k}^{I},\phantom{\rule{3.33333pt}{0ex}}F\left({x}_{k}\right)\rangle \le -\eta {\parallel F\left({x}_{k}\right)\parallel}^{2}\hfill \\ \langle {d}_{k}^{II},\phantom{\rule{3.33333pt}{0ex}}F\left({x}_{k}\right)\rangle \le -\delta {\parallel F\left({x}_{k}\right)\parallel}^{2}.\hfill \end{array}\right.$$
- (ii)
- The two search directions ${d}_{k}^{I}$ and ${d}_{k}^{II}$ satisfy the following inequalities$$\left\{\begin{array}{c}\eta \parallel F\left({x}_{k}\right)\parallel \le \parallel {d}_{k}^{I}\parallel \le \mu \parallel F\left({x}_{k}\right)\parallel \hfill \\ \delta \parallel F\left({x}_{k}\right)\parallel \le \parallel {d}_{k}^{II}\parallel \le \gamma \parallel F\left({x}_{k}\right)\parallel .\hfill \end{array}\right.$$

**Remark**

**2.**

- (i)
- We claim that there exists a step-size ${\beta}_{k}$ satisfying the line search (1) for any $k\ge 0.$ Suppose on the contrary that there exists some ${k}_{0}$ such that for any $i=0,1,2,...,$ the line search (1) is not satisfied, that is$$-\langle F({x}_{{k}_{0}}+\kappa {\varrho}^{i}d\left({w}_{{k}_{0}}\right)),\phantom{\rule{3.33333pt}{0ex}}d\left({w}_{{k}_{0}}\right)\rangle <\sigma \kappa {\varrho}^{i}\parallel d\left({w}_{{k}_{0}}\right){\parallel}^{2}{\parallel F({x}_{{k}_{0}}+\kappa {\varrho}^{i}d\left({w}_{{k}_{0}}\right))\parallel}^{1/c}.$$Since F is continuous and ${\lambda}_{k}^{II}$ is bounded for all k, letting $i\to \infty $ yields$$\parallel F\left({x}_{{k}_{0}}\right)\parallel \le 0.$$It is clear that the inequality (29) cannot hold. Hence the line search (1) is well-defined.
- (ii)
- The line search defined by (1) is more general than that of Reference [24].
- (iii)
- It follows from (15) and Assumption 1 that $\underset{k\to \infty}{lim}\parallel {w}_{k}-{x}_{k}\parallel =0.$

Algorithm 1: Two-Step Spectral Gradient Projection Method (TSSP) |

**Lemma**

**2.**

**Proof**

**of**

**Lemma**

**2.**

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**1.**

## 3. Numerical Results and Comparison

- (i)
- Spectral gradient projection method for monotone nonlinear equations with convex constraints proposed by Yu et al. [17].
- (ii)
- Two spectral gradient projection methods for constrained equations and their linear convergence rate proposed by Liu and Duan [25]. This method has two algorithms i.e., Algorithm 2.1 and Algorithm 2.2. We only compare our proposed method with Algorithm 2.1 since Algorithm 2.2 is similar with that Yu et al. [17].

**First experiment**. This experiment discusses the role of the parameter c in the definition of the line search (1) with regards to the performance of the TSSP algorithm. We solved all the test Problems 1–6 with dimension $n=1000,$ using all the given initial guesses in Table 1 by varying the values of c. That is, $c=\{1,2,3,4,5\}.$ The comparison is based on ITER, FVAL and norm of the objective function, (NORM), where the experimental results are presented in Table 2. CPU time results are omitted in Table 2 because virtually all are less than 1 s. The results obtained reveal that the parameter c slightly affected the performance of TSSP algorithm when solving Problems 2 and 6. For problem 2, Algorithm 1 TSSP recorded least ITER and FVAL when $c=4$ and 5 while different ITER and FVAL values recorded for different values of c may be associated with the random starting points chosen independently by MATLAB. However, extensive numerical experiment is needed to investigate the role of the parameter c in the performance of the TSSP algorithm.

**Second experiment**. This experiment presents the computational advantage of the proposed method in comparison with the two existing methods mentioned above based on ITER, FVAL and TIME. All the test problems $1\u20136$ were solved using the starting points in Table 1 with three (3) different dimensions $n=1000,$ 50,000 and 100,000. In this experiment, we take $c=2.$ The results obtained by each solver are reported in Table 3 and Table 4. The NORM results presented in Table 3 and Table 4 show that each solver successfully obtained solutions of all the test Problems 1–6. However, it is clear that the TSSP algorithm obtained the solutions of virtually all the test problems with least ITER, FVAL and TIME. These information are summarized in Figure 1, Figure 2 and Figure 3 based on the Dolan and Mor$\stackrel{\xb4}{e}$ performance profile [26]. This performance profile tells the percentage win by each solver. In all the experiments, we see from Figure 1, Figure 2 and Figure 3 that the proposed TSSP algorithm performs better with higher percentage win based on ITER, FVAL and TIME for solving all the test problems. In fact, the TSSP algorithm recorded 100 percent least FVAL for all the test problems.

**Problem**

**1**

**.**

**Problem**

**2**

**.**

**Problem**

**3**

**.**

**Problem**

**4**

**.**

**Problem**

**5**

**.**

**Problem**

**6**

**.**

## 4. Applications in Image Deblurring

#### Image Deblurring Experiment

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Dolan and Mor$\stackrel{\xb4}{e}$ performance profile with respect to number of iterations.

**Figure 2.**Dolan and Mor$\stackrel{\xb4}{e}$ performance profile with respect to number of function evaluation.

**Figure 4.**The original images (first row), the blurred images (second row), the restored images by methods TSSP (third row) and SGCS (last row).

Starting Points (SP) | Values |
---|---|

${x}_{1}$ | ${(0.1,0.1,0.1,\cdots ,0.1)}^{T}$ |

${x}_{2}$ | ${(\frac{1}{2},\frac{1}{{2}^{2}},\frac{1}{{2}^{3}},\cdots ,\frac{1}{{2}^{n}})}^{T}$ |

${x}_{3}$ | ${(2,2,\dots ,2)}^{T}$ |

${x}_{4}$ | ${(1,\frac{1}{2},\frac{1}{3},\cdots ,\frac{1}{n})}^{T}$ |

${x}_{5}$ | ${(1-\frac{1}{n},1-\frac{2}{n},1-\frac{3}{n},\cdots ,0)}^{T}$ |

${x}_{6}$ | rand$(0,1)$ |

c | 1 | 2 | 3 | 4 | 5 | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

SP | ITER | FVAL | NORM | ITER | FVAL | NORM | ITER | FVAL | NORM | ITER | FVAL | NORM | ITER | FVAL | NORM | |

P1 | ${x}_{1}$ | 4 | 6 | 2.18 × ${10}^{-7}$ | 4 | 6 | 2.18 × ${10}^{-7}$ | 4 | 6 | 2.18 × ${10}^{-7}$ | 4 | 6 | 2.18 × ${10}^{-7}$ | 4 | 6 | 2.18 × ${10}^{-7}$ |

${x}_{2}$ | 7 | 9 | 2.82 × ${10}^{-7}$ | 7 | 9 | 2.82 × ${10}^{-7}$ | 7 | 9 | 2.82 × ${10}^{-7}$ | 7 | 9 | 2.82 × ${10}^{-7}$ | 7 | 9 | 2.82 × ${10}^{-7}$ | |

${x}_{3}$ | 5 | 7 | 1.05 × ${10}^{-7}$ | 5 | 7 | 1.05 × ${10}^{-7}$ | 5 | 7 | 1.05 × ${10}^{-7}$ | 5 | 7 | 1.05 × ${10}^{-7}$ | 5 | 7 | 1.05 × ${10}^{-7}$ | |

${x}_{4}$ | 7 | 9 | 6.34 × ${10}^{-8}$ | 7 | 9 | 6.34 × ${10}^{-8}$ | 7 | 9 | 6.34 × ${10}^{-8}$ | 7 | 9 | 6.34 × ${10}^{-8}$ | 7 | 9 | 6.34 × ${10}^{-8}$ | |

${x}_{5}$ | 7 | 9 | 6.13 × ${10}^{-8}$ | 7 | 9 | 6.13 × ${10}^{-8}$ | 7 | 9 | 6.13 × ${10}^{-8}$ | 7 | 9 | 6.13 × ${10}^{-8}$ | 7 | 9 | 6.13 × ${10}^{-8}$ | |

${x}_{6}$ | 7 | 9 | 2.77 × ${10}^{-8}$ | 7 | 9 | 1.94 × ${10}^{-8}$ | 7 | 9 | 2.3 × ${10}^{-8}$ | 7 | 9 | 3.63 × ${10}^{-8}$ | 7 | 9 | 1.94 × ${10}^{-8}$ | |

P2 | ${x}_{1}$ | 3 | 5 | 2.41 × ${10}^{-8}$ | 3 | 5 | 2.41 × ${10}^{-8}$ | 3 | 5 | 2.41 × ${10}^{-8}$ | 3 | 5 | 2.41 × ${10}^{-8}$ | 3 | 5 | 2.41 × ${10}^{-8}$ |

${x}_{2}$ | 8 | 10 | 5.09 × ${10}^{-8}$ | 8 | 10 | 5.09 × ${10}^{-8}$ | 8 | 10 | 5.09 × ${10}^{-8}$ | 8 | 10 | 5.09 × ${10}^{-8}$ | 8 | 10 | 5.09 × ${10}^{-8}$ | |

${x}_{3}$ | 8 | 10 | 1.23 × ${10}^{-7}$ | 8 | 10 | 1.23 × ${10}^{-7}$ | 8 | 10 | 1.23 × ${10}^{-7}$ | 8 | 10 | 1.23 × ${10}^{-7}$ | 9 | 11 | 1.49 × ${10}^{-7}$ | |

${x}_{4}$ | 9 | 11 | 1.53 × ${10}^{-7}$ | 9 | 11 | 1.53 × ${10}^{-7}$ | 9 | 11 | 1.53 × ${10}^{-7}$ | 9 | 11 | 1.53 × ${10}^{-7}$ | 9 | 11 | 1.53 × ${10}^{-7}$ | |

${x}_{5}$ | 10 | 12 | 5.81 × ${10}^{-8}$ | 10 | 12 | 5.81 × ${10}^{-8}$ | 10 | 12 | 5.81 × ${10}^{-8}$ | 9 | 11 | 3.45 × ${10}^{-7}$ | 9 | 11 | 3.45 × ${10}^{-7}$ | |

${x}_{6}$ | 10 | 12 | 4.97 × ${10}^{-8}$ | 10 | 12 | 6.04 × ${10}^{-8}$ | 10 | 12 | 5.27 × ${10}^{-8}$ | 9 | 11 | 3.51 × ${10}^{-7}$ | 9 | 11 | 3.49 × ${10}^{-7}$ | |

P3 | ${x}_{1}$ | 3 | 5 | 4.03 × ${10}^{-8}$ | 3 | 5 | 4.03 × ${10}^{-8}$ | 3 | 5 | 4.03 × ${10}^{-8}$ | 3 | 5 | 4.03 × ${10}^{-8}$ | 3 | 5 | 4.03 × ${10}^{-8}$ |

${x}_{2}$ | 3 | 5 | 1.19 × ${10}^{-7}$ | 3 | 5 | 1.19 × ${10}^{-7}$ | 3 | 5 | 1.19 × ${10}^{-7}$ | 3 | 5 | 1.19 × ${10}^{-7}$ | 3 | 5 | 1.19 × ${10}^{-7}$ | |

${x}_{3}$ | 4 | 6 | 8.27 × ${10}^{-7}$ | 4 | 6 | 8.27 × ${10}^{-7}$ | 4 | 6 | 8.27 × ${10}^{-7}$ | 4 | 6 | 8.27 × ${10}^{-7}$ | 4 | 6 | 8.27 × ${10}^{-7}$ | |

${x}_{4}$ | 4 | 6 | 2.12 × ${10}^{-8}$ | 4 | 6 | 2.12 × ${10}^{-8}$ | 4 | 6 | 2.12 × ${10}^{-8}$ | 4 | 6 | 2.12 × ${10}^{-8}$ | 4 | 6 | 2.12 × ${10}^{-8}$ | |

${x}_{5}$ | 5 | 7 | 1.46 × ${10}^{-7}$ | 5 | 7 | 1.46 × ${10}^{-7}$ | 5 | 7 | 1.46 × ${10}^{-7}$ | 5 | 7 | 1.46 × ${10}^{-7}$ | 5 | 7 | 1.46 × ${10}^{-7}$ | |

${x}_{6}$ | 5 | 7 | 1.46 × ${10}^{-7}$ | 5 | 7 | 1.44 × ${10}^{-7}$ | 5 | 7 | 9.49 × ${10}^{-8}$ | 5 | 7 | 1.37 × ${10}^{-7}$ | 5 | 7 | 1.88 × ${10}^{-7}$ | |

P4 | ${x}_{1}$ | 1 | 2 | 0 | 1 | 2 | 0 | 1 | 2 | 0 | 1 | 2 | 0 | 1 | 2 | 0 |

${x}_{2}$ | 1 | 2 | 0 | 1 | 2 | 0 | 1 | 2 | 0 | 1 | 2 | 0 | 1 | 2 | 0 | |

${x}_{3}$ | 1 | 2 | 0 | 1 | 2 | 0 | 1 | 2 | 0 | 1 | 2 | 0 | 1 | 2 | 0 | |

${x}_{4}$ | 1 | 2 | 0 | 1 | 2 | 0 | 1 | 2 | 0 | 1 | 2 | 0 | 1 | 2 | 0 | |

${x}_{5}$ | 1 | 2 | 0 | 1 | 2 | 0 | 1 | 2 | 0 | 1 | 2 | 0 | 1 | 2 | 0 | |

${x}_{6}$ | 1 | 2 | 0 | 1 | 2 | 0 | 1 | 2 | 0 | 1 | 2 | 0 | 1 | 2 | 0 | |

P5 | ${x}_{1}$ | 3 | 5 | 7.96 × ${10}^{-7}$ | 3 | 5 | 7.96 × ${10}^{-7}$ | 3 | 5 | 7.96 × ${10}^{-7}$ | 3 | 5 | 7.96 × ${10}^{-7}$ | 3 | 5 | 7.96 × ${10}^{-7}$ |

${x}_{2}$ | 3 | 5 | 8.26 × ${10}^{-7}$ | 3 | 5 | 8.26 × ${10}^{-7}$ | 3 | 5 | 8.26 × ${10}^{-7}$ | 3 | 5 | 8.26 × ${10}^{-7}$ | 3 | 5 | 8.26 × ${10}^{-7}$ | |

${x}_{3}$ | 3 | 5 | 2.18 × ${10}^{-7}$ | 3 | 5 | 2.18 × ${10}^{-7}$ | 3 | 5 | 2.18 × ${10}^{-7}$ | 3 | 5 | 2.18 × ${10}^{-7}$ | 3 | 5 | 2.18 × ${10}^{-7}$ | |

${x}_{4}$ | 3 | 5 | 8.24 × ${10}^{-7}$ | 3 | 5 | 8.24 × ${10}^{-7}$ | 3 | 5 | 8.24 × ${10}^{-7}$ | 3 | 5 | 8.24 × ${10}^{-7}$ | 3 | 5 | 8.24 × ${10}^{-7}$ | |

${x}_{5}$ | 3 | 5 | 6.8 × ${10}^{-7}$ | 3 | 5 | 6.8 × ${10}^{-7}$ | 3 | 5 | 6.8 × ${10}^{-7}$ | 3 | 5 | 6.8 × ${10}^{-7}$ | 3 | 5 | 6.8 × ${10}^{-7}$ | |

${x}_{6}$ | 3 | 5 | 6.77 × ${10}^{-7}$ | 3 | 5 | 6.81 × ${10}^{-7}$ | 3 | 5 | 6.8 × ${10}^{-7}$ | 3 | 5 | 6.79 × ${10}^{-7}$ | 3 | 5 | 6.87 × ${10}^{-7}$ | |

P6 | ${x}_{1}$ | 3 | 5 | 1.07 × ${10}^{-7}$ | 3 | 5 | 1.07 × ${10}^{-7}$ | 3 | 5 | 1.07 × ${10}^{-7}$ | 3 | 5 | 1.07 × ${10}^{-7}$ | 3 | 5 | 1.07 × ${10}^{-7}$ |

${x}_{2}$ | 9 | 11 | 8.56 × ${10}^{-9}$ | 9 | 11 | 8.56 × ${10}^{-9}$ | 8 | 10 | 7.89 × ${10}^{-7}$ | 8 | 10 | 7.89 × ${10}^{-7}$ | 8 | 10 | 7.89 × ${10}^{-7}$ | |

${x}_{3}$ | 4 | 6 | 1.15 × ${10}^{-8}$ | 4 | 6 | 1.15 × ${10}^{-8}$ | 4 | 6 | 1.15 × ${10}^{-8}$ | 4 | 6 | 1.15 × ${10}^{-8}$ | 4 | 6 | 1.15 × ${10}^{-8}$ | |

${x}_{4}$ | 11 | 13 | 1.26 × ${10}^{-7}$ | 11 | 13 | 1.26 × ${10}^{-7}$ | 11 | 13 | 1.26 × ${10}^{-7}$ | 10 | 12 | 8.08 × ${10}^{-8}$ | 10 | 12 | 8.08 × ${10}^{-8}$ | |

${x}_{5}$ | 10 | 12 | 9.99 × ${10}^{-9}$ | 10 | 12 | 9.99 × ${10}^{-9}$ | 10 | 12 | 9.99 × ${10}^{-9}$ | 10 | 12 | 9.99 × ${10}^{-9}$ | 10 | 12 | 9.99 × ${10}^{-9}$ | |

${x}_{6}$ | 10 | 12 | 2.72 × ${10}^{-8}$ | 10 | 12 | 1.67 × ${10}^{-8}$ | 9 | 11 | 1.46 × ${10}^{-7}$ | 10 | 12 | 2.84 × ${10}^{-8}$ | 10 | 12 | 3.82 × ${10}^{-8}$ |

TSSP | SGPM | TSGP | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Problem | DIM | SP | ITER | FVAL | TIME | NORM | ITER | FVAL | TIME | NORM | ITER | FVAL | TIME | NORM |

P1 | 1000 | ${x}_{1}$ | 4 | 6 | 0.23423 | 2.18 × ${10}^{-7}$ | 5 | 11 | 0.071452 | 1.98 × ${10}^{-7}$ | 10 | 21 | 0.027483 | 4.81 × ${10}^{-7}$ |

${x}_{2}$ | 7 | 9 | 0.024129 | 2.82 × ${10}^{-7}$ | 21 | 43 | 0.020705 | 8.07 × ${10}^{-7}$ | 16 | 33 | 0.00993 | 7.7 × ${10}^{-7}$ | ||

${x}_{3}$ | 5 | 7 | 0.00479 | 1.05 × ${10}^{-7}$ | 8 | 17 | 0.005439 | 9.35 × ${10}^{-9}$ | 14 | 29 | 0.006688 | 7.24 × ${10}^{-7}$ | ||

${x}_{4}$ | 7 | 9 | 0.006348 | 6.34 × ${10}^{-8}$ | 22 | 45 | 0.007646 | 6.66 × ${10}^{-7}$ | 10 | 21 | 0.006682 | 8.79 × ${10}^{-7}$ | ||

${x}_{5}$ | 7 | 9 | 0.007179 | 6.13 × ${10}^{-8}$ | 7 | 15 | 0.005533 | 2.11 × ${10}^{-8}$ | 8 | 17 | 0.007977 | 9.36 × ${10}^{-7}$ | ||

${x}_{6}$ | 7 | 9 | 0.01028 | 3.35 × ${10}^{-8}$ | 7 | 15 | 0.008546 | 2.31 × ${10}^{-8}$ | 14 | 29 | 0.006145 | 6.95 × ${10}^{-7}$ | ||

50,000 | ${x}_{1}$ | 4 | 6 | 0.065863 | 6.63 × ${10}^{-8}$ | 5 | 11 | 0.06618 | 3.18 × ${10}^{-7}$ | 11 | 23 | 0.26178 | 8.02 × ${10}^{-7}$ | |

${x}_{2}$ | 7 | 9 | 0.06825 | 2.82 × ${10}^{-7}$ | 21 | 43 | 0.16873 | 8.07 × ${10}^{-7}$ | 16 | 33 | 0.23724 | 7.7 × ${10}^{-7}$ | ||

${x}_{3}$ | 5 | 7 | 0.1091 | 7.38 × ${10}^{-7}$ | 8 | 17 | 0.092683 | 6.61 × ${10}^{-8}$ | 16 | 33 | 0.19828 | 7.44 × ${10}^{-7}$ | ||

${x}_{4}$ | 7 | 9 | 0.07815 | 6.35 × ${10}^{-8}$ | 22 | 45 | 0.19587 | 6.67 × ${10}^{-7}$ | 10 | 21 | 0.11777 | 8.72 × ${10}^{-7}$ | ||

${x}_{5}$ | 8 | 10 | 0.090836 | 8 × ${10}^{-9}$ | 7 | 15 | 0.052803 | 1.34 × ${10}^{-7}$ | 17 | 35 | 0.212 | 7.38 × ${10}^{-7}$ | ||

${x}_{6}$ | 8 | 10 | 0.081964 | 6.53 × ${10}^{-9}$ | 7 | 15 | 0.049061 | 1.32 × ${10}^{-7}$ | 16 | 33 | 0.17428 | 8.9 × ${10}^{-7}$ | ||

100,000 | ${x}_{1}$ | 4 | 6 | 0.10587 | 7.46 × ${10}^{-8}$ | 5 | 11 | 0.12233 | 4.34 × ${10}^{-7}$ | 12 | 25 | 0.35698 | 4.49 × ${10}^{-7}$ | |

${x}_{2}$ | 7 | 9 | 0.13938 | 2.82 × ${10}^{-7}$ | 21 | 43 | 0.36669 | 8.07 × ${10}^{-7}$ | 16 | 33 | 0.53382 | 7.7 × ${10}^{-7}$ | ||

${x}_{3}$ | 6 | 8 | 0.17202 | 5.19 × ${10}^{-9}$ | 8 | 17 | 0.15426 | 9.34 × ${10}^{-8}$ | 17 | 35 | 0.58949 | 4.19 × ${10}^{-7}$ | ||

${x}_{4}$ | 7 | 9 | 0.15223 | 6.35 × ${10}^{-8}$ | 22 | 45 | 0.473 | 6.67 × ${10}^{-7}$ | 10 | 21 | 0.24947 | 8.72 × ${10}^{-7}$ | ||

${x}_{5}$ | 8 | 10 | 0.22073 | 1.13 × ${10}^{-8}$ | 7 | 15 | 0.1036 | 1.9 × ${10}^{-7}$ | 18 | 37 | 0.46019 | 4.18 × ${10}^{-7}$ | ||

${x}_{6}$ | 8 | 10 | 0.15584 | 1.18 × ${10}^{-8}$ | 7 | 15 | 0.1035 | 1.94 × ${10}^{-7}$ | 17 | 35 | 0.51277 | 7.07 × ${10}^{-7}$ | ||

P2 | 1000 | ${x}_{1}$ | 3 | 5 | 0.053692 | 2.41 × ${10}^{-8}$ | 19 | 39 | 0.022091 | 6.53 × ${10}^{-7}$ | 14 | 29 | 0.009273 | 9.17 × ${10}^{-7}$ |

${x}_{2}$ | 8 | 10 | 0.006247 | 5.09 × ${10}^{-8}$ | 19 | 39 | 0.008377 | 5.84 × ${10}^{-7}$ | 14 | 29 | 0.01022 | 7.19 × ${10}^{-7}$ | ||

${x}_{3}$ | 8 | 10 | 0.007091 | 1.23 × ${10}^{-7}$ | 24 | 49 | 0.009118 | 7.53 × ${10}^{-7}$ | 19 | 39 | 0.0171 | 5.49 × ${10}^{-7}$ | ||

${x}_{4}$ | 9 | 11 | 0.006962 | 1.53 × ${10}^{-7}$ | 20 | 41 | 0.00765 | 7.01 × ${10}^{-7}$ | 15 | 31 | 0.011734 | 7.44 × ${10}^{-7}$ | ||

${x}_{5}$ | 10 | 12 | 0.010053 | 5.81 × ${10}^{-8}$ | 23 | 47 | 0.010398 | 9.51 × ${10}^{-7}$ | 17 | 35 | 0.012062 | 9.65 × ${10}^{-7}$ | ||

${x}_{6}$ | 10 | 12 | 0.00733 | 7.06 × ${10}^{-8}$ | 23 | 47 | 0.009659 | 9.64 × ${10}^{-7}$ | 17 | 35 | 0.015027 | 9.61 × ${10}^{-7}$ | ||

50,000 | ${x}_{1}$ | 3 | 5 | 0.058732 | 1.71 × ${10}^{-7}$ | 22 | 45 | 0.27095 | 1.14 × ${10}^{-8}$ | 17 | 35 | 0.26277 | 4.05 × ${10}^{-7}$ | |

${x}_{2}$ | 8 | 10 | 0.1112 | 5.09 × ${10}^{-8}$ | 19 | 39 | 0.17143 | 5.86 × ${10}^{-7}$ | 14 | 29 | 0.2504 | 7.22 × ${10}^{-7}$ | ||

${x}_{3}$ | 8 | 10 | 0.11157 | 8.68 × ${10}^{-7}$ | 27 | 55 | 0.33645 | 1.35 × ${10}^{-8}$ | 21 | 43 | 0.44285 | 6.23 × ${10}^{-7}$ | ||

${x}_{4}$ | 9 | 11 | 0.11286 | 1.52 × ${10}^{-7}$ | 20 | 41 | 0.18758 | 7.05 × ${10}^{-7}$ | 15 | 31 | 0.26827 | 7.49 × ${10}^{-7}$ | ||

${x}_{5}$ | 10 | 12 | 0.13738 | 4.2 × ${10}^{-7}$ | 26 | 53 | 0.24905 | 8.68 × ${10}^{-7}$ | 20 | 41 | 0.32368 | 4.36 × ${10}^{-7}$ | ||

${x}_{6}$ | 10 | 12 | 0.15618 | 4.3 × ${10}^{-7}$ | 26 | 53 | 0.36605 | 8.69 × ${10}^{-7}$ | 20 | 41 | 0.31156 | 4.35 × ${10}^{-7}$ | ||

100,000 | ${x}_{1}$ | 3 | 5 | 0.091277 | 2.42 × ${10}^{-7}$ | 21 | 43 | 0.41274 | 3.2 × ${10}^{-8}$ | 17 | 35 | 0.55481 | 5.73 × ${10}^{-7}$ | |

${x}_{2}$ | 8 | 10 | 0.19525 | 5.09 × ${10}^{-8}$ | 19 | 39 | 0.35045 | 5.86 × ${10}^{-7}$ | 14 | 29 | 0.48308 | 7.22 × ${10}^{-7}$ | ||

${x}_{3}$ | 9 | 11 | 0.27631 | 1.21 × ${10}^{-8}$ | 27 | 55 | 0.628 | 1.91 × ${10}^{-8}$ | 21 | 43 | 0.66514 | 8.81 × ${10}^{-7}$ | ||

${x}_{4}$ | 9 | 11 | 0.32755 | 1.52 × ${10}^{-7}$ | 20 | 41 | 0.44064 | 7.05 × ${10}^{-7}$ | 15 | 31 | 0.48698 | 7.49 × ${10}^{-7}$ | ||

${x}_{5}$ | 10 | 12 | 0.25909 | 5.94 × ${10}^{-7}$ | 27 | 55 | 0.54056 | 6.2 × ${10}^{-7}$ | 20 | 41 | 0.74411 | 6.16 × ${10}^{-7}$ | ||

${x}_{6}$ | 10 | 12 | 0.28941 | 5.64 × ${10}^{-7}$ | 27 | 55 | 0.57984 | 6.19 × ${10}^{-7}$ | 20 | 41 | 0.66926 | 6.16 × ${10}^{-7}$ | ||

P3 | 1000 | ${x}_{1}$ | 3 | 5 | 0.027505 | 4.03 × ${10}^{-8}$ | 5 | 11 | 0.004418 | 1.97 × ${10}^{-8}$ | 11 | 23 | 0.004932 | 4.32 × ${10}^{-7}$ |

${x}_{2}$ | 3 | 5 | 0.002243 | 1.19 × ${10}^{-7}$ | 5 | 11 | 0.004881 | 3.84 × ${10}^{-8}$ | 8 | 17 | 0.006148 | 5.84 × ${10}^{-7}$ | ||

${x}_{3}$ | 4 | 6 | 0.003901 | 8.27 × ${10}^{-7}$ | 6 | 13 | 0.00328 | 4.62 × ${10}^{-7}$ | 13 | 27 | 0.010201 | 6.63 × ${10}^{-7}$ | ||

${x}_{4}$ | 4 | 6 | 0.002999 | 2.12 × ${10}^{-8}$ | 5 | 11 | 0.002216 | 3.93 × ${10}^{-7}$ | 11 | 23 | 0.007859 | 9.11 × ${10}^{-7}$ | ||

${x}_{5}$ | 5 | 7 | 0.004183 | 1.46 × ${10}^{-7}$ | 6 | 13 | 0.002305 | 4.67 × ${10}^{-8}$ | 14 | 29 | 0.009221 | 8.68 × ${10}^{-7}$ | ||

${x}_{6}$ | 5 | 7 | 0.002517 | 1.08 × ${10}^{-7}$ | 6 | 13 | 0.002634 | 4.62 × ${10}^{-8}$ | 14 | 29 | 0.007964 | 8.55 × ${10}^{-7}$ | ||

50,000 | ${x}_{1}$ | 3 | 5 | 0.037839 | 2.85 × ${10}^{-7}$ | 5 | 11 | 0.046813 | 1.39 × ${10}^{-7}$ | 13 | 27 | 0.3024 | 4.88 × ${10}^{-7}$ | |

${x}_{2}$ | 3 | 5 | 0.043702 | 1.19 × ${10}^{-7}$ | 5 | 11 | 0.075547 | 3.84 × ${10}^{-8}$ | 8 | 17 | 0.13 | 5.84 × ${10}^{-7}$ | ||

${x}_{3}$ | 5 | 7 | 0.067654 | 5.79 × ${10}^{-8}$ | 7 | 15 | 0.053226 | 3.24 × ${10}^{-8}$ | 15 | 31 | 0.2087 | 7.48 × ${10}^{-7}$ | ||

${x}_{4}$ | 4 | 6 | 0.043413 | 2.13 × ${10}^{-8}$ | 5 | 11 | 0.041688 | 3.93 × ${10}^{-7}$ | 11 | 23 | 0.20256 | 9.12 × ${10}^{-7}$ | ||

${x}_{5}$ | 6 | 8 | 0.05719 | 1.03 × ${10}^{-8}$ | 6 | 13 | 0.052196 | 3.31 × ${10}^{-7}$ | 16 | 33 | 0.31984 | 9.81 × ${10}^{-7}$ | ||

${x}_{6}$ | 6 | 8 | 0.080035 | 1.01 × ${10}^{-8}$ | 6 | 13 | 0.047676 | 3.31 × ${10}^{-7}$ | 16 | 33 | 0.31002 | 9.75 × ${10}^{-7}$ | ||

100,000 | ${x}_{1}$ | 3 | 5 | 0.074948 | 4.03 × ${10}^{-7}$ | 5 | 11 | 0.095503 | 1.97 × ${10}^{-7}$ | 13 | 27 | 0.38401 | 6.89 × ${10}^{-7}$ | |

${x}_{2}$ | 3 | 5 | 0.068017 | 1.19 × ${10}^{-7}$ | 5 | 11 | 0.070957 | 3.84 × ${10}^{-8}$ | 8 | 17 | 0.18947 | 5.84 × ${10}^{-7}$ | ||

${x}_{3}$ | 5 | 7 | 0.26113 | 8.19 × ${10}^{-8}$ | 7 | 15 | 0.13652 | 4.58 × ${10}^{-8}$ | 16 | 33 | 0.51768 | 4.22 × ${10}^{-7}$ | ||

${x}_{4}$ | 4 | 6 | 0.13668 | 2.13 × ${10}^{-8}$ | 5 | 11 | 0.073903 | 3.93 × ${10}^{-7}$ | 11 | 23 | 0.31674 | 9.12 × ${10}^{-7}$ | ||

${x}_{5}$ | 6 | 8 | 0.16527 | 1.46 × ${10}^{-8}$ | 6 | 13 | 0.080572 | 4.68 × ${10}^{-7}$ | 17 | 35 | 0.46576 | 5.54 × ${10}^{-7}$ | ||

${x}_{6}$ | 6 | 8 | 0.15469 | 1.5 × ${10}^{-8}$ | 6 | 13 | 0.1035 | 4.68 × ${10}^{-7}$ | 17 | 35 | 0.49075 | 5.52 × ${10}^{-7}$ |

TSSP | SGPM | TSGP | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Problem | DIM | SP | ITER | FVAL | TIME | NORM | ITER | FVAL | TIME | NORM | ITER | FVAL | TIME | NORM |

P4 | 1000 | ${x}_{1}$ | 1 | 2 | 0.024999 | 0 | 1 | 3 | 0.002082 | 0 | 1 | 3 | 0.002665 | 0 |

${x}_{2}$ | 1 | 2 | 0.00153 | 0 | 1 | 3 | 0.001492 | 0 | 10 | 21 | 0.013708 | 6.89 × ${10}^{-7}$ | ||

${x}_{3}$ | 1 | 2 | 0.002033 | 0 | 1 | 3 | 0.001752 | 0 | 1 | 3 | 0.003047 | 0 | ||

${x}_{4}$ | 1 | 2 | 0.001605 | 0 | 11 | 23 | 0.00416 | 8.04 × ${10}^{-7}$ | 1 | 3 | 0.001204 | 0 | ||

${x}_{5}$ | 1 | 2 | 0.001898 | 0 | 20 | 41 | 0.006966 | 6.33 × ${10}^{-7}$ | 17 | 35 | 0.024513 | 5.22 × ${10}^{-7}$ | ||

${x}_{6}$ | 1 | 2 | 0.002199 | 0 | 20 | 41 | 0.005064 | 6.89 × ${10}^{-7}$ | 17 | 35 | 0.009568 | 5.82 × ${10}^{-7}$ | ||

50,000 | ${x}_{1}$ | 1 | 2 | 0.009816 | 0 | 1 | 3 | 0.02162 | 0 | 1 | 3 | 0.027567 | 0 | |

${x}_{2}$ | 1 | 2 | 0.015792 | 0 | 1 | 3 | 0.010298 | 0 | 10 | 21 | 0.11697 | 6.89 × ${10}^{-7}$ | ||

${x}_{3}$ | 1 | 2 | 0.019539 | 0 | 1 | 3 | 0.009652 | 0 | 1 | 3 | 0.04943 | 0 | ||

${x}_{4}$ | 1 | 2 | 0.022933 | 0 | 11 | 23 | 0.087566 | 7.32 × ${10}^{-7}$ | 1 | 3 | 0.016433 | 0 | ||

${x}_{5}$ | 1 | 2 | 0.014115 | 0 | 23 | 47 | 0.15224 | 5.8 × ${10}^{-7}$ | 19 | 39 | 0.25956 | 5.87 × ${10}^{-7}$ | ||

${x}_{6}$ | 1 | 2 | 0.015431 | 0 | 23 | 47 | 0.17212 | 5.81 × ${10}^{-7}$ | 19 | 39 | 0.34396 | 5.8 × ${10}^{-7}$ | ||

100,000 | ${x}_{1}$ | 1 | 2 | 0.029333 | 0 | 1 | 3 | 0.018075 | 0 | 1 | 3 | 0.027167 | 0 | |

${x}_{2}$ | 1 | 2 | 0.029502 | 0 | 1 | 3 | 0.021229 | 0 | 10 | 21 | 0.20962 | 6.89 × ${10}^{-7}$ | ||

${x}_{3}$ | 1 | 2 | 0.029413 | 0 | 1 | 3 | 0.029106 | 0 | 1 | 3 | 0.042574 | 0 | ||

${x}_{4}$ | 1 | 2 | 0.029978 | 0 | 11 | 23 | 0.15785 | 7.32 × ${10}^{-7}$ | 1 | 3 | 0.029647 | 0 | ||

${x}_{5}$ | 1 | 2 | 0.024201 | 0 | 23 | 47 | 0.37503 | 8.2 × ${10}^{-7}$ | 19 | 39 | 0.46268 | 8.31 × ${10}^{-7}$ | ||

${x}_{6}$ | 1 | 2 | 0.023344 | 0 | 23 | 47 | 0.30268 | 8.28 × ${10}^{-7}$ | 19 | 39 | 0.44689 | 8.24 × ${10}^{-7}$ | ||

P5 | 1000 | ${x}_{1}$ | 3 | 5 | 0.026584 | 7.96 × ${10}^{-7}$ | 22 | 45 | 0.012781 | 4.76 × ${10}^{-7}$ | 20 | 41 | 0.02354 | 8.85 × ${10}^{-7}$ |

${x}_{2}$ | 3 | 5 | 0.003513 | 8.26 × ${10}^{-7}$ | 21 | 43 | 0.021446 | 9.78 × ${10}^{-7}$ | 20 | 41 | 0.026418 | 9.18 × ${10}^{-7}$ | ||

${x}_{3}$ | 3 | 5 | 0.003116 | 2.18 × ${10}^{-7}$ | 20 | 41 | 0.011601 | 5.12 × ${10}^{-7}$ | 19 | 39 | 0.02013 | 6.08 × ${10}^{-7}$ | ||

${x}_{4}$ | 3 | 5 | 0.002986 | 8.24 × ${10}^{-7}$ | 24 | 49 | 0.008455 | 1.26 × ${10}^{-7}$ | 20 | 41 | 0.038547 | 9.16 × ${10}^{-7}$ | ||

${x}_{5}$ | 3 | 5 | 0.003386 | 6.8 × ${10}^{-7}$ | 26 | 53 | 0.015125 | 2.64 × ${10}^{-8}$ | 20 | 41 | 0.021084 | 7.56 × ${10}^{-7}$ | ||

${x}_{6}$ | 3 | 5 | 0.003238 | 6.8 × ${10}^{-7}$ | 26 | 53 | 0.014889 | 2.67 × ${10}^{-8}$ | 20 | 41 | 0.019351 | 7.56 × ${10}^{-7}$ | ||

50,000 | ${x}_{1}$ | 6 | 8 | 0.10841 | 7.24 × ${10}^{-7}$ | 18 | 37 | 0.28111 | 1.99 × ${10}^{-8}$ | 22 | 45 | 0.63837 | 9.98 × ${10}^{-7}$ | |

${x}_{2}$ | 6 | 8 | 0.12008 | 7.52 × ${10}^{-7}$ | 19 | 39 | 0.35252 | 5.32 × ${10}^{-7}$ | 23 | 47 | 0.59083 | 4.14 × ${10}^{-7}$ | ||

${x}_{3}$ | 4 | 6 | 0.091808 | 7.79 × ${10}^{-7}$ | 17 | 35 | 0.24494 | 5.52 × ${10}^{-7}$ | 21 | 43 | 0.61762 | 6.85 × ${10}^{-7}$ | ||

${x}_{4}$ | 6 | 8 | 0.10356 | 7.52 × ${10}^{-7}$ | 19 | 39 | 0.26199 | 1.04 × ${10}^{-8}$ | 23 | 47 | 0.58276 | 4.14 × ${10}^{-7}$ | ||

${x}_{5}$ | 6 | 8 | 0.1118 | 6.19 × ${10}^{-7}$ | 22 | 45 | 0.42086 | 5.64 × ${10}^{-8}$ | 22 | 45 | 0.68866 | 8.53 × ${10}^{-7}$ | ||

${x}_{6}$ | 6 | 8 | 0.13042 | 6.19 × ${10}^{-7}$ | 20 | 41 | 0.30457 | 2.21 × ${10}^{-7}$ | 22 | 45 | 0.57108 | 8.53 × ${10}^{-7}$ | ||

100,000 | ${x}_{1}$ | 7 | 9 | 0.3991 | 5.17 × ${10}^{-7}$ | 17 | 35 | 0.53229 | 5.58 × ${10}^{-8}$ | 23 | 47 | 1.4183 | 5.64 × ${10}^{-7}$ | |

${x}_{2}$ | 7 | 9 | 0.33016 | 5.37 × ${10}^{-7}$ | 19 | 39 | 0.70642 | 7.53 × ${10}^{-7}$ | 23 | 47 | 1.2936 | 5.85 × ${10}^{-7}$ | ||

${x}_{3}$ | 5 | 7 | 0.22423 | 5.57 × ${10}^{-7}$ | 17 | 35 | 0.53979 | 7.8 × ${10}^{-7}$ | 21 | 43 | 1.0728 | 9.69 × ${10}^{-7}$ | ||

${x}_{4}$ | 7 | 9 | 0.31926 | 5.37 × ${10}^{-7}$ | 18 | 37 | 0.59493 | 2.92 × ${10}^{-8}$ | 23 | 47 | 1.3459 | 5.85 × ${10}^{-7}$ | ||

${x}_{5}$ | 6 | 8 | 0.25384 | 8.75 × ${10}^{-7}$ | 19 | 39 | 0.69354 | 1.21 × ${10}^{-8}$ | 23 | 47 | 1.2682 | 4.82 × ${10}^{-7}$ | ||

${x}_{6}$ | 6 | 8 | 0.37171 | 8.75 × ${10}^{-7}$ | 20 | 41 | 0.62058 | 3.13 × ${10}^{-7}$ | 23 | 47 | 1.3007 | 4.82 × ${10}^{-7}$ | ||

P6 | 1000 | ${x}_{1}$ | 3 | 5 | 0.010955 | 1.07 × ${10}^{-7}$ | 23 | 47 | 0.01257 | 6.93 × ${10}^{-7}$ | 19 | 39 | 0.015919 | 4.69 × ${10}^{-7}$ |

${x}_{2}$ | 9 | 11 | 0.006 | 8.56 × ${10}^{-9}$ | 23 | 47 | 0.012384 | 9.68 × ${10}^{-7}$ | 19 | 39 | 0.01716 | 5.74 × ${10}^{-7}$ | ||

${x}_{3}$ | 4 | 6 | 0.006474 | 1.15 × ${10}^{-8}$ | 6 | 13 | 0.002199 | 2.21 × ${10}^{-7}$ | 13 | 27 | 0.009428 | 8.36 × ${10}^{-7}$ | ||

${x}_{4}$ | 11 | 13 | 0.008502 | 1.26 × ${10}^{-7}$ | 23 | 47 | 0.013556 | 9.78 × ${10}^{-7}$ | 19 | 39 | 0.012728 | 5.69 × ${10}^{-7}$ | ||

${x}_{5}$ | 10 | 12 | 0.010102 | 9.99 × ${10}^{-9}$ | 7 | 15 | 0.004183 | 2.36 × ${10}^{-7}$ | 19 | 39 | 0.010011 | 4.42 × ${10}^{-7}$ | ||

${x}_{6}$ | 10 | 12 | 0.007011 | 7.27 × ${10}^{-9}$ | 7 | 15 | 0.006891 | 2.93 × ${10}^{-7}$ | 19 | 39 | 0.009542 | 4.06 × ${10}^{-7}$ | ||

50,000 | ${x}_{1}$ | 3 | 5 | 0.061255 | 7.6 × ${10}^{-7}$ | 24 | 49 | 0.23097 | 2.6 × ${10}^{-8}$ | 21 | 43 | 0.32775 | 5.28 × ${10}^{-7}$ | |

${x}_{2}$ | 11 | 13 | 0.13565 | 2.66 × ${10}^{-7}$ | 25 | 51 | 0.23841 | 1.83 × ${10}^{-8}$ | 21 | 43 | 0.34316 | 6.46 × ${10}^{-7}$ | ||

${x}_{3}$ | 4 | 6 | 0.062462 | 8.1 × ${10}^{-8}$ | 7 | 15 | 0.065855 | 8.32 × ${10}^{-9}$ | 15 | 31 | 0.25529 | 9.4 × ${10}^{-7}$ | ||

${x}_{4}$ | 14 | 16 | 0.20446 | 1.62 × ${10}^{-8}$ | 26 | 53 | 0.26012 | 8.69 × ${10}^{-7}$ | 21 | 43 | 0.31348 | 6.46 × ${10}^{-7}$ | ||

${x}_{5}$ | 10 | 12 | 0.14657 | 7.62 × ${10}^{-8}$ | 8 | 17 | 0.071141 | 8.94 × ${10}^{-9}$ | 21 | 43 | 0.34669 | 5.05 × ${10}^{-7}$ | ||

${x}_{6}$ | 10 | 12 | 0.15597 | 7.06 × ${10}^{-8}$ | 8 | 17 | 0.091229 | 8.84 × ${10}^{-9}$ | 21 | 43 | 0.35675 | 5.27 × ${10}^{-7}$ | ||

100,000 | ${x}_{1}$ | 4 | 6 | 0.11184 | 5.71 × ${10}^{-9}$ | 24 | 49 | 0.57769 | 3.68 × ${10}^{-8}$ | 21 | 43 | 0.62764 | 7.47 × ${10}^{-7}$ | |

${x}_{2}$ | 14 | 16 | 0.44177 | 1.79 × ${10}^{-7}$ | 25 | 51 | 0.45188 | 2.59 × ${10}^{-8}$ | 21 | 43 | 0.6427 | 9.14 × ${10}^{-7}$ | ||

${x}_{3}$ | 4 | 6 | 0.20305 | 1.15 × ${10}^{-7}$ | 7 | 15 | 0.14039 | 1.18 × ${10}^{-8}$ | 16 | 33 | 0.51687 | 5.3 × ${10}^{-7}$ | ||

${x}_{4}$ | 14 | 16 | 0.30629 | 1.42 × ${10}^{-7}$ | 27 | 55 | 0.49475 | 6.18 × ${10}^{-7}$ | 21 | 43 | 0.65503 | 9.14 × ${10}^{-7}$ | ||

${x}_{5}$ | 9 | 11 | 0.18947 | 2.07 × ${10}^{-8}$ | 8 | 17 | 0.13416 | 1.26 × ${10}^{-8}$ | 21 | 43 | 0.67669 | 7.15 × ${10}^{-7}$ | ||

${x}_{6}$ | 9 | 11 | 0.25267 | 2.25 × ${10}^{-8}$ | 8 | 17 | 0.16038 | 1.25 × ${10}^{-8}$ | 21 | 43 | 0.67886 | 7.17 × ${10}^{-7}$ |

TSSP | SGCS | ||||||||
---|---|---|---|---|---|---|---|---|---|

Image | Size | ITER | TIME(s) | SNR | SSIM | ITER | TIME(s) | SNR | SSIM |

Lena | 256 × 256 | 113 | 8.84 | 24.25 | 0.90 | 218 | 11.09 | 23.70 | 0.90 |

House | 256 × 256 | 121 | 16.53 | 22.86 | 0.87 | 235 | 17.47 | 23.61 | 0.87 |

Pepper | 256 × 256 | 100 | 8.69 | 27.58 | 0.89 | 167 | 12.05 | 27.02 | 0.89 |

Camera man | 256 × 256 | 21 | 2.20 | 20.33 | 0.84 | 28 | 2.19 | 20.56 | 0.84 |

Barbara | 512 × 512 | 22 | 12.69 | 19.16 | 0.76 | 23 | 11.08 | 19.16 | 0.76 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Awwal, A.M.; Wang, L.; Kumam, P.; Mohammad, H.
A Two-Step Spectral Gradient Projection Method for System of Nonlinear Monotone Equations and Image Deblurring Problems. *Symmetry* **2020**, *12*, 874.
https://doi.org/10.3390/sym12060874

**AMA Style**

Awwal AM, Wang L, Kumam P, Mohammad H.
A Two-Step Spectral Gradient Projection Method for System of Nonlinear Monotone Equations and Image Deblurring Problems. *Symmetry*. 2020; 12(6):874.
https://doi.org/10.3390/sym12060874

**Chicago/Turabian Style**

Awwal, Aliyu Muhammed, Lin Wang, Poom Kumam, and Hassan Mohammad.
2020. "A Two-Step Spectral Gradient Projection Method for System of Nonlinear Monotone Equations and Image Deblurring Problems" *Symmetry* 12, no. 6: 874.
https://doi.org/10.3390/sym12060874