Next Article in Journal
On the Geometry of Semi-Invariant Submanifolds in (α, p)-Golden Riemannian Manifolds
Previous Article in Journal
Practical Security of Continuous Variable Measurement- Device-Independent Quantum Key Distribution with Local Local Oscillator
 
 
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

Enhanced Projection Method for the Solution of the System of Nonlinear Equations Under a More General Assumption than Pseudo-Monotonicity and Lipschitz Continuity

by
Kanikar Muangchoo
1 and
Auwal Bala Abubakar
2,3,4,*
1
Department of Mathematics and Statistics, Faculty of Science and Technology, Rajamangala University of Technology Phra Nakhon (RMUTP), 1381 Pracharat 1 Road, Wongsawang, Bang Sue, Bangkok 10800, Thailand
2
Department of Art and Science, George Mason University, Songdomunhwa-ro 119-4, Yeonsu-gu, Incheon 21985, Republic of Korea
3
Numerical Optimization Research Group, Department of Mathematical Sciences, Faculty of Physical Sciences, Bayero University, Kano 700241, Nigeria
4
Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, Medunsa, Pretoria 0204, South Africa
*
Author to whom correspondence should be addressed.
Mathematics 2024, 12(23), 3734; https://doi.org/10.3390/math12233734
Submission received: 7 November 2024 / Revised: 22 November 2024 / Accepted: 26 November 2024 / Published: 27 November 2024

Abstract

:
In this manuscript, we propose an efficient algorithm for solving a class of nonlinear operator equations. The algorithm is an improved version of previously established method. The algorithm’s features are as follows: (i) the search direction is bounded and satisfies the sufficient descent condition; (ii) the global convergence is achieved when the operator is continuous and satisfies a condition weaker than pseudo-monotonicity. Moreover, by comparing it with previously established method the algorithm’s efficiency was shown. The comparison was based on the iteration number required for each algorithm to solve a particular problem and the time taken. Some benchmark test problems, which included monotone and pseudo-monotone problems, were considered for the experiments. Lastly, the algorithm was utilized to solve the logistic regression (prediction) model.

1. Introduction

In the field of computational mathematics, solving nonlinear equations stands as a fundamental challenge with diverse applications cutting across areas such as optimization, physics, engineering, and economics. The investigation for efficient and robust numerical methods to address this challenge has led to the development of numerous algorithms. Among these, the conjugate gradient (CG) method appears to be a powerful tool, particularly due to its convergence properties and effectiveness in solving large-scale nonlinear systems.
Consider finding the solution of
F ( v ) = 0 ,
where F : R n R n is continuous. There are a lot of CG algorithms for solving (1) in the literature, most of which require F to be monotone and Lipschitz continuous (see, for example, [1,2,3,4,5,6,7] and references therein). Recently algorithms that require F to be pseudo-monotone, which is weaker than monotone, were proposed (see [8,9] for more details). In particular, Liu et al. [8] proposed an efficient CG projection algorithm for solving (1). The operator F under consideration was assumed to be pseudo-monotone and continuous, which are weaker than monotonicity and Lipschitz continuity, respectively. In addition, the CG parameter consists of a fixed constant c. After careful observations of the algorithm by Liu et al. [8], we posed the following questions: (i) Can we impose a weaker assumption than pseudo-monotone on F and still achieve convergence? (ii) Can we replace the fixed constant c with an adaptive parameter and achieve a more efficient algorithm?
The answers to the questions above are given in the next section. In what follows, we give a summary of the contributions:
1.
The article proposes an efficient algorithm for solving (1).
2.
The operator F is assumed to satisfy a weaker condition than pseudo-monotonicity.
3.
The proposed algorithm is globally convergent.
4.
On some test problems, which include monotone and pseudo-monotone, the proposed algorithm performs better than that of Liu et al. [8].
This manuscript is organized as follows: Section 2 consists of the proposed algorithm and its convergence analysis. Section 3 captures the numerical experiments, where the performance of the proposed method is observed for some benchmark test problems. In Section 4, we apply the proposed algorithm to solve prediction models. Section 5 has the conclusion of this work.

2. The Proposed Algorithm and Its Convergence

In this section, we start by recalling the search direction of the iterative algorithm proposed by Liu et al. in [8] called Algorithm 2.1. The search direction is defined as
d k = F ( v k ) if k = 0 F ( v k ) + β k d k 1 , elsewhere ,
where
β k : = c F ( v k ) T d k 1 d k 1 T w k 1 , c [ 0 , 1 ) ,
w k 1 = y k 1 + δ k 1 d k 1 , y k 1 = F ( v k ) F ( v k 1 ) ,
 and 
δ k 1 = 1 + max 0 , d k 1 T y k 1 d k 1 2 .
As noted in [8], the direction becomes the steepest descent when c = 0 or F ( v k ) T d k 1 = 0 .
After careful observation, we realized that replacing the constant c in (3) with an adaptive parameter with the same domain as c can enhance the performance of the algorithm. As such, we propose the following parameter
θ k : = 1 ( F ( v k ) T d k 1 ) 2 F ( v k ) 2 d k 1 2 ,
to replace c.
Remark 1. 
Observe that θ k [ 0 , 1 ) since by Cauchy-Schwartz inequality, we have that
F ( v k ) T d k 1 F ( v k ) d k 1 .
Hence, we use the following for β k :
β k : = θ k F ( v k ) T d k 1 d k 1 T w k 1 .
Remark 2. 
Applying the definition of w k 1 and δ k 1 from (4) and (5), respectively, we have
d k 1 T w k 1 d k 1 T y k 1 + d k 1 2 d k 1 T y k 1 = d k 1 2 > 0 .
Definition 1. 
A function F is pseudo-monotone, if for all v 1 , v 2 R n
F ( v 1 ) T ( v 2 v 1 ) 0 implies F ( v 2 ) T ( v 2 v 1 ) 0 ,
Before stating the proposed algorithm, we state the following assumptions:
Assumption 1. 
The solution set of (1) is non-empty.
Assumption 2. 
F is continuous and for any v such that F ( v ) = 0 and v R n , then
F ( v ) T ( v v ) 0 .
Remark 3. 
Observe that if F satisfies (9), then F satisfies (10). However, the converse is not true.
By Remark 3, a function that satisfies (10) is more general than a pseudo-monotone function, which is more general than a monotone function.
In what follows, we present the proposed Algorithm 1 for solving (1) under Assumptions 1 and 2.
Algorithm 1: (Modified Algorithm 2.1)
Initialization.  v 0 R n , ρ ( 0 , 1 ) , λ ( 0 , 2 ) , σ ( 0 , 1 ) , ϵ > 0 . Set k = 0 .
Step 0. Check if F ( v k ) ϵ and stop. Else, compute d k by Equations (2) and (7).
Step 1. Evaluate u k = v k + α k d k , where the step size α k = ρ i with i being the smallest non-negative integer satisfying
F ( u k ) T d k σ α k F ( u k ) d k 2 .
Step 2. Check if F ( u k ) ϵ , and stop. Else, evaluate
v k + 1 = v k λ F ( v k ) T ( v k u k ) F ( u k ) 2 F ( u k ) .
Step 3. Set k = k + 1 and repeat the procedure.
We omit the proof of the following Lemma as it is similar to the one in [8].
Lemma 1 
([8]). For all k and θ k [ 0 , 1 ) , the search direction defined by Equation (2) satisfies the inequalities
d k T F ( v k ) ( 1 θ k ) F ( v k ) 2 ,
and
d k ( 1 + θ k ) F ( v k ) .
Lemma 2. 
Suppose v R n such that F ( v ) = 0 and { v k } generated by the  Modified Algorithm 2.1, then
v k + 1 v 2 v k v 2 λ ( 2 λ ) σ 2 v k u k 4 .
Proof. 
By (11), and (10),
F ( u k ) T ( v k v ) = F ( u k ) T ( v k v u k + u k ) = F ( u k ) T ( v k u k ) + F ( u k ) T ( u k v ) F ( u k ) T ( v k u k ) σ α k 2 F ( u k ) d k 2 0 .
Now, by (11), (12), (15) and λ ( 0 , 2 ) , we have
v k + 1 v 2 = v k λ F ( u k ) T ( v k u k ) F ( u k ) 2 F ( u k ) v 2 = ( v k v ) + ( λ F ( u k ) T ( v k u k ) ) F ( u k ) 2 F ( u k ) ) 2 = ( v k v ) + ( λ F ( u k ) T ( v k u k ) ) F ( u k ) 2 F ( u k ) ) T ( v k v ) + ( λ F ( u k ) T ( v k u k ) ) F ( u k ) 2 F ( u k ) ) = ( v k v ) T ( v k v ) 2 λ F ( u k ) T ( v k u k ) F ( u k ) 2 F ( u k ) T ( v k v ) + λ 2 F ( u k ) T ( v k u k ) ) F ( u k ) 2 F ( u k ) T F ( u k ) T ( v k u k ) ) F ( u k ) 2 F ( u k ) = v k v 2 2 λ F ( u k ) T ( v k u k ) F ( u k ) 2 F ( u k ) T ( v k v ) + λ 2 ( F ( u k ) T ( v k u k ) ) 2 F ( u k ) 2 v k v 2 2 λ ( F ( u k ) T ( v k u k ) ) 2 F ( u k ) 2 + λ 2 ( F ( u k ) T ( v k u k ) ) 2 F ( u k ) 2 = v k v 2 λ ( 2 λ ) F ( u k ) T ( v k u k ) 2 F ( u k ) 2 v k v 2 λ ( 2 λ ) σ 2 α k 4 d k 4 v k v 2 .
   □
Remark 4. 
lim k v k v exists from (16) since { v k v } is non-increasing and hence convergent.
Remark 5. 
From Remark 4, { v k v } is bounded. That is, there exists a positive constant M ¯ such that for all k,
v k v M ¯ .
Applying the triangle inequality on (17), we have
v k M ¯ + v : = M ˜ .
Furthermore, { F ( v k ) } is bounded.
The proof of the following theorem can be found in [8].
Theorem 1 
([8]). If Assumptions 1 and 2 hold and { v k } is generated by Modified Algorithm 2.1, then
lim inf k F ( v k ) = 0 .
Corollary 1. 
The sequence { v k } generated by  Modified Algorithm 2.1converges to some v in the sense that F ( v ) = 0 .
Proof. 
The continuity of F implies that there is a limit point v of the sequence { v k } with F ( v ) = 0 , that is, v is a solution. In addition, because v k v converges from Remark 4, v is an accumulation point of v k v , then { v k } v .    □

3. Numerical Experiments

We consider two experiments in this section. Firstly, we perform experiments on the system of monotone equations since every monotone function is pseudo-monotone. Secondly, we consider pseudo-monotone functions that are not monotone.

3.1. Solving System of Monotone Equations

As every monotone function is pseudo-monotone, in this subsection, we will apply Modified Algorithm 2.1 to solve ten (10) systems of monotone equations given in Appendix A with different dimensions and starting points. To show the impact of the adaptive parameter θ k in the search direction of Modified Algorithm 2.1, we compared it with Algorithm 2.1 in [8], where the parameter is fixed. We consider the following for the comparison:
  • Five dimensions: n = 1000 , 5000 , 10 , 000 , 50 , 000 , 100 , 000
  • Seven initial points: v 0 = ( 1 8 , 1 8 , , 1 8 ) T ,   v 0 = 2 5 , 2 5 , 2 5 T ,   v 0 = ( 1 10 , 1 10 , , 1 10 ) T , v 0 = 1 100 , 1 100 , , 1 100 T ,   v 0 = ( 1 2 , 1 2 , 1 2 ) T ,   v 0 = 1 5 , 1 5 , , 1 5 T , v 0 = ( 1 4 , 1 4 , 1 4 ) T .
  • Parameters for Modified Algorithm 2.1: ρ = 0.8 , λ = 1.2 , σ = 10 4 , ϵ = 10 6 .
  • The choice of λ is by trial and error in the interval ( 0 , 2 ) and we got the best result at 1.7.
  • Parameters for Algorithm 2.1: As chosen in [8].
  • Stopping condition: when F ( v k ) 10 6 .
The results obtained from the experiment can be found in Appendix B where #ITN, #FEN, CPUTIMS, and Norm represent iteration number, function evaluation number, CPU time in seconds, and norm of the function at the solution, respectively. However, for clarity, we employ the performance profiles of Dolan and Moré [10]. It gives us a graphical interpretation of the percentage success for each algorithm (solver) based on the #ITN, #FEN, and CPUTIMS. The algorithm with the highest percentage on the y-axes of the plot is considered the best (see Figure 1, Figure 2 and Figure 3). MATLAB R2023b was used for the numerical experiments on a personal computer with 2.3 Ghz, 8 GB RAM, using Windows 7 operating system.
From Figure 1, Figure 2 and Figure 3, one can see that the percentage of Modified Algorithm 2.1 is higher than that of Algorithm 2.1 in all the figures. This is associated with an adaptive parameter θ k in Modified Algorithm 2.1 that replaces the fixed constant in Algorithm 2.1.

3.2. Solving Pseudo-Monotone Equations

As observed in Section 1, Liu et al. [8] proposed an efficient algorithm for solving a system of pseudo-monotone equations. However, they could only provide numerical examples for solving monotone equations. In this subsection, we will provide and solve two systems of equations where F is not monotone but pseudo-monotone, as shown in Table 1.
Consider the following pseudo-monotone equations defined by:
  P1: [11] F : R 2 R 2 defined as
F 1 ( v ) = ( v 1 + ( v 2 1 ) 2 ) ( 1 + v 2 ) F 2 ( v ) = v 1 3 v 1 ( v 2 1 ) 2 .
P2: [12] F : R 3 R 3 defined as
F ( v ) = ( exp v 2 + Φ ) M v , where , M = 1 0 1 0 1.5 0 1 0 2 , and Φ = 0.2 .
Table 1 shows that each algorithm successfully found the solution of problems P1 and P2. However, in all instances, Modified Algorithm 2.1 achieved the solution with fewer iterations and function evaluations than Algorithm 2.1. Moreover, the time taken for Algorithm 2.1 to attain a solution was less in six out of the ten instances than in Modified Algorithm 2.1. This may be as a result of the computational cost associated with computing the adaptive parameter in Modified Algorithm 2.1.

4. Solving a Prediction Model

In this section, we will apply Modified Algorithm 2.1 to solve a prediction model called logistic regression. The logistic regression model can be used to predict a categorical dependent variable, which makes it an important tool in machine learning. The logistic regression model [13] is a problem of the form
min v R n f ( v ) ,
where
f ( v ) = 1 N i = 1 N log ( 1 + exp c i l i T v ) + ξ 2 v 2 .
ξ is the regularization parameter, 1 N i = 1 N log ( 1 + exp c i l i T v ) is the loss function, c i { 1 , 1 } is the binary class and l i R n is the data variable. f is strongly convex and v is a unique solution of (19) if and only if it solves the nonlinear equation:
0 = F ( v ) : = 1 N i = 1 N c i ( exp c i l i T v ) l i 1 + exp c i l i T v + ξ v .
See [14,15] and references therein. Furthermore, Equation (21) is monotone, and therefore the Modified Algorithm 2.1 can solve (21).
As such, Modified Algorithm 2.1 was implemented using MATLAB R2023b to solve (21). Its performance was compared with the following algorithms; Algorithm 2.1 proposed by Liu et al. in [8] and DFMRMIL by Koorapetse et al. [16]. Leveraging the library for support vector machines called LIBSVM (https://www.csie.ntu.edu.tw/~cjlin/libsvmtools/datasets/ accessed on 1 July 2024) [17], ten test instances were considered (see Table 2). The starting point is chosen as v 0 = 4 × ( r a n d ( n , 1 ) 0.5 ) just like in [14] with ξ = 0.1 . The same parameters as in the last section were considered for Modified Algorithm 2.1, Algorithm 2.1, and DFMRMIL; the parameters were chosen as given in [8] and [16], respectively. In addition, each instance was replicated five times and the average values of #ITN, #FEN, CPUTIMS, and Norm were recorded (see Table 3). It can be seen from Table 3 that the mean values of #ITN, #FEN, CPUTIMS for Modified Algorithm 2.1 are less than those for Algorithm 2.1 and DFMRMIL in 80 % of the datasets.

5. Conclusions

An improved projection method based on the CG method for solving continuous nonlinear equations was proposed. The improvement was a result of an adaptive parameter θ k that replaces the fixed constant c in the CG parameter proposed by [8]. Moreover, the global convergence was established under a weaker assumption than pseudo-monotone on the operator F. Numerical experiments were performed on some benchmark nonlinear equations that involve monotone and pseudo-monotone operators. The results of the experiments indicate that the proposed algorithm is outstanding. This is surely a sign that replacing c with θ k was a better choice. Furthermore, the algorithm was employed to solve a prediction model and the results were also satisfactory. Finally, based on the reported results, we can conclude by recommending an adaptive parameter over a fixed constant.

Author Contributions

Conceptualization, A.B.A. and K.M.; methodology, A.B.A.; software, K.M.; validation, K.M. and A.B.A.; formal analysis, A.B.A.; investigation, K.M.; resources, K.M.; data curation, A.B.A.; writing—original draft preparation, A.B.A.; writing—review and editing, K.M.; visualization, K.M.; supervision, A.B.A.; project administration, K.M.; funding acquisition, A.B.A. and K.M. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

The data presented in this study are available in https://www.csie.ntu.edu.tw/~cjlin/libsvmtools/datasets/ (accessed on 1 July 2024). These data were derived from the following resources available in the public domain [17].

Acknowledgments

The first author was financially supported by Rajamangala University of Technology Phra Nakhon (RMUTP) Research Scholarship. The second author thank the Department of Mathematics and Applied Mathematics at the Sefako Makgatho Health Sciences University.

Conflicts of Interest

The authors declare no conflicts of interest.

Appendix A

Below are the benchmark test problems considered, where F : R n R n is F ( v ) = ( F 1 ( v ) , F 2 ( v ) , , F n ( v ) ) T and v = ( v 1 , v 2 , , v n ) T .
Prob. 1: Modified exponential function [18]
F 1 ( v ) = exp v 1 1 F i ( v ) = exp v i + v i 1 , i = 2 , 3 , , n .
Prob. 2: Logarithmic function [18]
F i ( v ) = log ( v i + 1 ) v i n , i = 1 , 2 , , n .
Prob. 3: [19]
F i ( v ) = 2 v i sin v i , i = 1 , 2 , 3 , , n .
Prob. 4: Discrete boundary value problem [18]
F 1 ( v ) = 2 v 1 + 0.5 q 2 ( v 1 + h ) 3 v 2 , F i ( v ) = 2 v i v i 1 + v i + 1 + 0.5 q 2 ( v i + i h ) 3 , for i = 2 , , n 1 , F n ( v ) = 2 v n v n 1 + 0.5 q 2 ( v n + n h ) 3 , q = 1 n + 1 .
Prob. 5: Exponential function [20]
F i ( v ) = exp v i 1 , i = 1 , 2 , , n .
Prob. 6: Tridiagonal exponential function [18]
F 1 ( v ) = v 1 exp cos ( q ( v 1 + v 2 ) ) F i ( v ) = v i exp cos ( q ( v i 1 + v i + v i + 1 ) ) , i = 2 , , n 1 , F n ( v ) = v n exp cos ( q ( v n 1 + v n ) ) , q = 1 n + 1 .
Prob. 7: Nonsmooth function [19]
F i ( v ) = v i sin | v i 1 | , i = 1 , 2 , 3 , , n .
Prob. 8: Modified Zhou and Li function [21]
F 1 ( v ) = 2 v 1 + sin v 1 1 , F i ( v ) = v i 1 + 2 v i + sin v i 1 , for i = 2 , , n 1 , F n ( v ) = 2 v n + sin v n 1 .
Prob. 9: Trig-exponential function [22]
F i ( v ) = e v i 2 + 3 sin v i cos v i 1 , i = 1 , 2 , , n .
Prob. 10: Pursuit-evasion problem [22]
F i ( v ) = 8 v i 1 , i = 1 , 2 , , n .

Appendix B

Table A1. Result for Prob. 1 and 2.
Table A1. Result for Prob. 1 and 2.
Modified Algorithm 2.1Algorithm 2.1
S/N#ITN#FENCPUTIMSNorm#ITN#FENCPUTIMSNorm
127320.0125496.59 × 10 7 43450.0070617.32 × 10 7
213170.0057474.03 × 10 7 26280.0063457.32 × 10 7
311150.0037889.32 × 10 7 26280.0060488.01 × 10 7
47110.0036346.81 × 10 7 20220.0042187.32 × 10 7
511150.0096588.75 × 10 7 32340.0094169.34 × 10 7
69130.0034976.65 × 10 7 25270.0048899.94 × 10 7
79130.002825.99 × 10 7 24260.004678.17 × 10 7
822270.0250315.75 × 10 7 41430.0368659.73 × 10 7
910140.0111023.62 × 10 7 24260.0187979.67 × 10 7
107110.0086118.68 × 10 7 25270.0214997.65 × 10 7
117110.0095829.61 × 10 7 18200.0154219.70 × 10 7
1211150.0124123.00 × 10 7 31330.0266588.92 × 10 7
139130.0118628.03 × 10 7 24260.0213469.52 × 10 7
149130.0106094.84 × 10 7 22240.0201039.24 × 10 7
1519240.0393399.80 × 10 7 41430.0608578.30 × 10 7
169130.0199018.73 × 10 7 24260.0328028.24 × 10 7
1710140.0197332.23 × 10 7 24260.0425719.02 × 10 7
187110.0141357.78 × 10 8 18200.0305668.28 × 10 7
1911150.0238045.03 × 10 7 31330.0480297.60 × 10 7
209130.0189872.19 × 10 7 24260.0371138.12 × 10 7
218120.0172716.79 × 10 7 22240.0341377.73 × 10 7
2216210.103334.12 × 10 7 40420.239287.94 × 10 7
238120.0717352.16 × 10 7 23250.125027.86 × 10 7
249130.0766534.27 × 10 7 23250.155528.62 × 10 7
259130.0732013.24 × 10 7 17190.110628.06 × 10 7
269130.0794457.61 × 10 8 30320.198927.26 × 10 7
2711150.0928266.60 × 10 7 23250.146787.75 × 10 7
2810140.0885758.78 × 10 7 21230.130647.28 × 10 7
2915200.252424.03 × 10 7 39410.442749.37 × 10 7
3010140.140221.67 × 10 7 22240.263919.28 × 10 7
3110140.158742.09 × 10 7 23250.281057.34 × 10 7
329130.136823.62 × 10 7 16180.20029.94 × 10 7
3310140.176278.06 × 10 7 29310.34238.56 × 10 7
3412160.206073.21 × 10 7 22240.242319.15 × 10 7
3512160.20322.07 × 10 7 20220.249918.58 × 10 7
36780.003355.16 × 10 8 10100.0062394.54 × 10 7
37670.0021094.29 × 10 8 10100.0028214.05 × 10 7
38670.0022054.73 × 10 8 10100.0029084.98 × 10 7
39880.0028097.51 × 10 7 990.0035012.27 × 10 7
40670.0024292.30 × 10 7 11110.0031392.18 × 10 7
41670.0022575.23 × 10 8 10100.0028987.47 × 10 7
42670.0021955.01 × 10 8 10100.002778.83 × 10 7
43780.0080971.05 × 10 7 11110.0097952.04 × 10 7
44670.0073688.75 × 10 8 10100.0107839.59 × 10 7
45670.0068139.65 × 10 8 11110.0125272.18 × 10 7
46560.0062923.49 × 10 7 990.0096535.33 × 10 7
47670.0073434.64 × 10 7 11110.011045.21 × 10 7
48670.007691.07 × 10 7 11110.0142843.27 × 10 7
49670.0075511.03 × 10 7 11110.0113533.87 × 10 7
50780.0151751.47 × 10 7 11110.0169712.91 × 10 7
51670.0124831.22 × 10 7 11110.0174222.53 × 10 7
52670.0146751.35 × 10 7 11110.0195073.11 × 10 7
53560.0131334.89 × 10 7 990.0143257.59 × 10 7
54670.0132936.48 × 10 7 11110.0174367.43 × 10 7
55670.0126271.49 × 10 7 11110.0182464.67 × 10 7
56670.0126241.43 × 10 7 11110.0184735.52 × 10 7
57780.0606623.25 × 10 7 11110.0655186.56 × 10 7
58670.0534212.71 × 10 7 11110.070275.69 × 10 7
59670.0472232.99 × 10 7 11110.0728417.00 × 10 7
60670.0450324.35 × 10 8 10100.068563.16 × 10 7
61780.0508135.74 × 10 8 12120.0794343.10 × 10 7
62670.0574123.30 × 10 7 12120.0856911.95 × 10 7
63670.0521063.18 × 10 7 12120.0796662.30 × 10 7
64780.0953434.59 × 10 7 11110.141689.29 × 10 7
65670.10413.83 × 10 7 11110.140938.05 × 10 7
66670.0965234.22 × 10 7 11110.150549.91 × 10 7
67670.11216.14 × 10 8 10100.136044.47 × 10 7
68780.111568.10 × 10 8 12120.162584.39 × 10 7
69670.10174.67 × 10 7 12120.170062.75 × 10 7
70670.105964.49 × 10 7 12120.153513.26 × 10 7
Table A2. Result for Prob. 3 and 4.
Table A2. Result for Prob. 3 and 4.
Modified Algorithm 2.1Algorithm 2.1
S/N#ITN#FENCPUTIMSNorm#ITN#FENCPUTIMSNorm
71670.0023043.45 × 10 7 16170.0076159.47 × 10 7
72560.0018653.11 × 10 7 17180.0038034.96 × 10 7
73560.0016033.79 × 10 7 17180.0037236.17 × 10 7
74450.0019028.09 × 10 7 14150.0031436.60 × 10 7
75450.0017975.03 × 10 7 18190.0041897.98 × 10 7
76560.0018175.44 × 10 7 17180.0040699.62 × 10 7
77560.0017416.08 × 10 7 18190.0039144.97 × 10 7
78670.0071137.71 × 10 7 17180.0125898.97 × 10 7
79560.0058266.95 × 10 7 18190.014714.70 × 10 7
80560.0054588.49 × 10 7 18190.0157355.84 × 10 7
81560.006037.24 × 10 8 15160.0133626.25 × 10 7
82560.0064194.50 × 10 8 19200.0164257.56 × 10 7
83670.0075734.87 × 10 8 18190.0156999.11 × 10 7
84670.0063835.44 × 10 8 19200.0177874.71 × 10 7
85780.0149224.36 × 10 8 18190.0251465.37 × 10 7
86560.0100379.83 × 10 7 18190.0267886.65 × 10 7
87670.0124954.80 × 10 8 18190.0262498.26 × 10 7
88560.0107121.02 × 10 7 15160.0261288.84 × 10 7
89560.0095916.36 × 10 8 20210.120914.53 × 10 7
90670.0128146.88 × 10 8 19200.0655155.46 × 10 7
91670.0097867.69 × 10 8 19200.0322076.66 × 10 7
92780.0589869.76 × 10 8 19200.124355.09 × 10 7
93670.0513128.79 × 10 8 19200.109186.30 × 10 7
94670.0510251.07 × 10 7 19200.106587.83 × 10 7
95560.0341372.29 × 10 7 16170.0995658.37 × 10 7
96560.0337631.42 × 10 7 21220.116814.29 × 10 7
97670.0537041.54 × 10 7 20210.113585.17 × 10 7
98670.044351.72 × 10 7 20210.114026.31 × 10 7
99780.0953471.38 × 10 7 19200.21947.20 × 10 7
100670.0800871.24 × 10 7 19200.211588.91 × 10 7
101670.107611.52 × 10 7 20210.219434.69 × 10 7
102560.0769393.24 × 10 7 17180.203735.01 × 10 7
103560.0840552.01 × 10 7 21220.231256.06 × 10 7
104670.0782712.18 × 10 7 20210.220967.31 × 10 7
105670.0949792.43 × 10 7 20210.22378.92 × 10 7
10632370.0203477.96 × 10 7 22240.0142476.06 × 10 7
10728330.0153876.01 × 10 7 18200.0091449.09 × 10 7
10828330.0166497.52 × 10 7 19210.0099635.72 × 10 7
10923280.0133647.52 × 10 7 15170.0083336.96 × 10 7
11031360.0172726.59 × 10 7 21230.0123086.01 × 10 7
11129340.0174747.23 × 10 7 19210.0086339.15 × 10 7
11229340.0177569.04 × 10 7 20220.0101315.79 × 10 7
11332370.0943688.13 × 10 7 22240.0397935.80 × 10 7
11428330.0821686.15 × 10 7 18200.0357678.78 × 10 7
11528330.0744687.68 × 10 7 19210.0388235.52 × 10 7
11623280.0536987.69 × 10 7 15170.0288477.00 × 10 7
11731360.0797696.74 × 10 7 21230.041585.71 × 10 7
11829340.0708967.39 × 10 7 19210.0383678.83 × 10 7
11929340.0660479.24 × 10 7 20220.0379475.55 × 10 7
12034390.14968.94 × 10 7 22240.0800095.83 × 10 7
12130350.132466.70 × 10 7 18200.077218.97 × 10 7
12230350.148898.38 × 10 7 19210.0895455.61 × 10 7
12325300.121798.42 × 10 7 15170.0628937.38 × 10 7
12433380.147337.35 × 10 7 21230.0887685.73 × 10 7
12531360.129068.12 × 10 7 19210.0801118.97 × 10 7
12632370.137396.16 × 10 7 20220.0832655.61 × 10 7
12733380.663099.95 × 10 7 22240.423086.45 × 10 7
12829340.590257.46 × 10 7 19210.322255.30 × 10 7
12929340.585519.33 × 10 7 19210.324226.63 × 10 7
13024290.506659.41 × 10 7 15170.264259.52 × 10 7
13132370.640348.18 × 10 7 21230.356086.45 × 10 7
13230350.597259.04 × 10 7 20220.348365.20 × 10 7
13331360.624196.86 × 10 7 20220.355366.50 × 10 7
13435401.42286.54 × 10 7 22240.795447.04 × 10 7
13530351.26198.22 × 10 7 19210.702355.96 × 10 7
13631361.27916.16 × 10 7 19210.686517.45 × 10 7
13726311.05656.12 × 10 7 16180.560925.27 × 10 7
13833381.36819.05 × 10 7 21230.754477.12 × 10 7
13931361.28199.85 × 10 7 20220.726875.80 × 10 7
14032371.2947.40 × 10 7 20220.709127.25 × 10 7
Table A3. Result for Prob. 5 and 6.
Table A3. Result for Prob. 5 and 6.
Modified Algorithm 2.1Algorithm 2.1
S/N#ITN#FENCPUTIMSNorm#ITN#FENCPUTIMSNorm
14117200.0049374.52 × 10 7 21220.0080099.50 × 10 7
142560.0017137.90 × 10 8 16170.0032458.03 × 10 7
143560.0016492.37 × 10 7 16170.003128.87 × 10 7
144450.0015217.09 × 10 7 14150.0028746.38 × 10 7
1459110.0029734.71 × 10 7 18190.0036757.72 × 10 7
146670.0018044.71 × 10 8 16170.0030998.59 × 10 7
147670.0017529.45 × 10 8 16170.0031635.99 × 10 7
14818210.0167773.90 × 10 7 22230.0148038.99 × 10 7
149560.0047761.77 × 10 7 17180.0121747.61 × 10 7
150560.0046895.29 × 10 7 17180.0100718.40 × 10 7
151560.0048536.34 × 10 8 15160.0110696.04 × 10 7
15210120.0104222.44 × 10 7 19200.0133937.31 × 10 7
153670.006211.05 × 10 7 17180.0117748.14 × 10 7
154670.0054772.11 × 10 7 17180.0122735.68 × 10 7
15518210.0289695.51 × 10 7 23240.0252675.39 × 10 7
156560.0085432.50 × 10 7 18190.0218784.56 × 10 7
157560.0079357.48 × 10 7 18190.0202625.03 × 10 7
158560.0076358.97 × 10 8 15160.0186538.55 × 10 7
15910120.0141683.46 × 10 7 20210.0242334.38 × 10 7
160670.0093461.49 × 10 7 18190.0218544.87 × 10 7
161670.0093032.99 × 10 7 17180.0198618.03 × 10 7
16219220.106794.75 × 10 7 24250.109545.10 × 10 7
163560.0248695.59 × 10 7 19200.0864884.32 × 10 7
164670.0331296.69 × 10 8 19200.0981194.77 × 10 7
165560.0315832.01 × 10 7 16170.0799648.10 × 10 7
16610120.0670597.73 × 10 7 20210.0902979.79 × 10 7
167670.0433453.33 × 10 7 19200.0865134.62 × 10 7
168670.0399996.69 × 10 7 18190.0804097.60 × 10 7
16919220.211916.72 × 10 7 24250.223547.22 × 10 7
170560.0526727.90 × 10 7 19200.192056.11 × 10 7
171670.0600959.46 × 10 8 19200.176496.74 × 10 7
172560.0494322.84 × 10 7 17180.15564.85 × 10 7
17311130.111652.54 × 10 7 21220.194985.87 × 10 7
174670.0622954.71 × 10 7 19200.176836.53 × 10 7
175670.0693499.45 × 10 7 19200.173974.55 × 10 7
176670.0032742.21 × 10 7 20210.00996.54 × 10 7
177670.0025653.36 × 10 7 20210.0064169.96 × 10 7
178670.0025423.33 × 10 7 20210.0058179.86 × 10 7
179670.0025213.48 × 10 7 21220.0077364.36 × 10 7
180670.0027292.85 × 10 7 20210.0060598.44 × 10 7
181670.0025523.23 × 10 7 20210.0061059.58 × 10 7
182670.0025973.17 × 10 7 20210.0059129.39 × 10 7
183670.0085484.97 × 10 7 21220.0249776.20 × 10 7
184670.0100677.58 × 10 7 21220.0249489.45 × 10 7
185670.0101347.51 × 10 7 21220.0271879.36 × 10 7
186670.0084477.84 × 10 7 21220.0230519.78 × 10 7
187670.0091626.42 × 10 7 21220.0245168.01 × 10 7
188670.0102277.29 × 10 7 21220.0272049.09 × 10 7
189670.0096057.15 × 10 7 21220.0279128.91 × 10 7
190670.0152647.04 × 10 7 21220.0479748.78 × 10 7
191780.0178834.29 × 10 8 22230.0487595.66 × 10 7
192780.0215654.25 × 10 8 22230.0572825.61 × 10 7
193780.0179474.44 × 10 8 22230.0511615.86 × 10 7
194670.0159289.09 × 10 7 22230.0496984.80 × 10 7
195780.0202414.13 × 10 8 22230.0449665.45 × 10 7
196780.0202344.04 × 10 8 22230.053255.34 × 10 7
197780.080976.30 × 10 8 22230.234078.31 × 10 7
198780.0773869.59 × 10 8 23240.251485.37 × 10 7
199780.087569.50 × 10 8 23240.226745.31 × 10 7
200780.0874849.92 × 10 8 23240.238485.55 × 10 7
201780.0856988.13 × 10 8 23240.232794.55 × 10 7
202780.0793539.23 × 10 8 23240.230785.16 × 10 7
203780.0761349.04 × 10 8 23240.241525.06 × 10 7
204780.165168.90 × 10 8 23240.525924.98 × 10 7
205780.173421.36 × 10 7 23240.515097.59 × 10 7
206780.180531.34 × 10 7 23240.567987.52 × 10 7
207780.189721.40 × 10 7 23240.512327.85 × 10 7
208780.196491.15 × 10 7 23240.48766.43 × 10 7
209780.178321.30 × 10 7 23240.481847.30 × 10 7
210780.160831.28 × 10 7 23240.478767.15 × 10 7
Table A4. Result for Prob. 7 and 8.
Table A4. Result for Prob. 7 and 8.
Modified Algorithm 2.1Algorithm 2.1
S/N#ITN#FENCPUTIMSNorm#ITN#FENCPUTIMSNorm
2117110.0027872.87 × 10 7 18200.0101184.63 × 10 7
2129120.0028334.88 × 10 7 13140.003442.32 × 10 7
2139120.0027584.87 × 10 7 12130.0028839.67 × 10 7
2149120.0032254.50 × 10 7 13140.0030522.69 × 10 7
2158110.002751.71 × 10 7 14160.0034155.73 × 10 7
2169120.0029334.56 × 10 7 12130.002827.96 × 10 7
2179120.0028564.13 × 10 7 12130.0027456.72 × 10 7
2187110.0083876.41 × 10 7 19210.015444.17 × 10 7
21910130.0104571.64 × 10 7 13140.0148475.18 × 10 7
22010130.010251.64 × 10 7 13140.0118554.91 × 10 7
22110130.0103611.51 × 10 7 13140.0145546.03 × 10 7
2228110.0098853.82 × 10 7 15170.0144165.16 × 10 7
22310130.0114751.53 × 10 7 13140.0108244.05 × 10 7
2249120.0099219.23 × 10 7 13140.0104053.41 × 10 7
2257110.0156219.07 × 10 7 19210.0249185.89 × 10 7
22610130.0178742.32 × 10 7 13140.0198267.33 × 10 7
22710130.0178792.32 × 10 7 13140.019446.95 × 10 7
22810130.0169852.14 × 10 7 13140.0172648.52 × 10 7
2298110.0170645.41 × 10 7 15170.0204747.30 × 10 7
23010130.0211752.17 × 10 7 13140.0203045.72 × 10 7
23110130.0226811.96 × 10 7 13140.0181444.83 × 10 7
2328120.0742861.62 × 10 7 20220.12965.31 × 10 7
23310130.0862615.19 × 10 7 14150.0743733.72 × 10 7
23410130.0851895.18 × 10 7 14150.0755563.53 × 10 7
23510130.0702524.79 × 10 7 14150.0976564.33 × 10 7
2369120.0619711.82 × 10 7 16180.110446.57 × 10 7
23710130.0725584.85 × 10 7 14150.0780552.91 × 10 7
23810130.0722854.39 × 10 7 14150.0855652.45 × 10 7
2398120.13032.29 × 10 7 20220.242787.51 × 10 7
24010130.136227.33 × 10 7 14150.141115.26 × 10 7
24110130.131847.32 × 10 7 14150.167254.99 × 10 7
24210130.153746.77 × 10 7 14150.156776.12 × 10 7
2439120.122032.57 × 10 7 16180.196779.29 × 10 7
24410130.129916.85 × 10 7 14150.169134.11 × 10 7
24510130.138986.21 × 10 7 14150.174883.47 × 10 7
24647530.0137438.15 × 10 7 37390.0137149.08 × 10 7
24751570.0137716.94 × 10 7 34360.00826.34 × 10 7
24849550.0131748.89 × 10 7 33350.008439.02 × 10 7
24943490.0120999.14 × 10 7 34360.0079699.82 × 10 7
25039450.010618.97 × 10 7 24270.0061478.48 × 10 7
25150560.0130179.10 × 10 7 32340.0074228.19 × 10 7
25248540.013517.91 × 10 7 31330.0074736.91 × 10 7
25349550.0485049.24 × 10 7 38400.0301858.37 × 10 7
25451570.0516239.04 × 10 7 34360.0286717.98 × 10 7
25552580.0555277.82 × 10 7 33350.0318177.71 × 10 7
25644500.0466677.33 × 10 7 35370.0306146.94 × 10 7
25740460.0479118.29 × 10 7 25280.0246838.52 × 10 7
25850560.0520696.95 × 10 7 33350.0314796.86 × 10 7
25951570.0579437.07 × 10 7 31330.0320969.41 × 10 7
26049550.0931729.56 × 10 7 37390.0633999.06 × 10 7
26153590.105437.07 × 10 7 33350.0587668.70 × 10 7
26252580.0948788.57 × 10 7 33350.0569389.10 × 10 7
26345510.0998418.27 × 10 7 35370.0590725.78 × 10 7
26442480.0778747.18 × 10 7 16190.0319777.17 × 10 7
26549550.104489.72 × 10 7 32340.0571668.37 × 10 7
26651570.0968767.82 × 10 7 32340.0522857.18 × 10 7
26748540.424029.25 × 10 7 44460.339998.69 × 10 7
26854600.483688.64 × 10 7 34360.261496.01 × 10 7
26952580.472918.54 × 10 7 34360.254766.15 × 10 7
27045510.399628.20 × 10 7 35370.282839.76 × 10 7
27146520.411437.04 × 10 7 20230.155339.29 × 10 7
27253590.468598.28 × 10 7 33350.262065.90 × 10 7
27352580.464649.22 × 10 7 32340.231488.85 × 10 7
27449550.948458.90 × 10 7 38400.577326.68 × 10 7
27554600.973989.02 × 10 7 35370.589259.39 × 10 7
27651570.948238.51 × 10 7 35370.566876.82 × 10 7
27745510.817969.20 × 10 7 34360.678879.27 × 10 7
27845510.819298.11 × 10 7 29310.512348.85 × 10 7
27952580.971687.44 × 10 7 34360.510237.71 × 10 7
28051570.965479.76 × 10 7 32340.544929.94 × 10 7
Table A5. Result for Prob. 9 and 10.
Table A5. Result for Prob. 9 and 10.
Modified Algorithm 2.1Algorithm 2.1
S/N#ITN#FENCPUTIMSNorm#ITN#FENCPUTIMSNorm
28110150.0061162.50 × 10 7 28320.0127319.50 × 10 7
2824120.0030651.52 × 10 7 7100.003448.84 × 10 8
2834120.0030111.85 × 10 7 7100.0028491.23 × 10 7
2843110.0028967.36 × 10 7 690.0024841.28 × 10 7
2859140.0052333.84 × 10 7 25260.0105517.49 × 10 7
2864120.0029811.11 × 10 8 7100.0026952.77 × 10 7
2874120.00346.46 × 10 7 7100.0027084.42 × 10 7
28810150.0214685.60 × 10 7 30340.0377546.19 × 10 7
2894120.0099833.41 × 10 7 7100.0106391.98 × 10 7
2904120.0117554.14 × 10 7 7100.0084662.75 × 10 7
2914120.0116761.09 × 10 8 690.0095632.85 × 10 7
2929140.0210638.59 × 10 7 26270.0397978.78 × 10 7
2934120.0116922.48 × 10 8 7100.0107796.20 × 10 7
2945130.011849.58 × 10 9 7100.0100629.87 × 10 7
29510150.0468327.92 × 10 7 30340.0662458.75 × 10 7
2964120.0195884.82 × 10 7 7100.0179932.79 × 10 7
2974120.0194495.85 × 10 7 7100.0173823.89 × 10 7
2984120.0184841.54 × 10 8 690.0153334.03 × 10 7
29910150.0429821.40 × 10 7 27280.0813666.51 × 10 7
3004120.0201163.51 × 10 8 7100.0191628.77 × 10 7
3015130.0221061.35 × 10 8 8110.0214516.77 × 10 8
30211160.186112.14 × 10 7 32360.309565.70 × 10 7
3035130.0852597.15 × 10 9 7100.0711416.25 × 10 7
3045130.0861188.68 × 10 9 7100.0706388.70 × 10 7
3054120.0666013.45 × 10 8 690.0682059.02 × 10 7
30610150.17913.13 × 10 7 28290.362337.63 × 10 7
3074120.0730267.84 × 10 8 8110.072439.52 × 10 8
3085130.0947813.03 × 10 8 8110.0764681.51 × 10 7
30911160.370983.02 × 10 7 32360.571268.06 × 10 7
3105130.161421.01 × 10 8 7100.141568.84 × 10 7
3115130.167261.23 × 10 8 8110.16275.97 × 10 8
3124120.153834.88 × 10 8 7100.145426.19 × 10 8
31310150.331814.42 × 10 7 29300.764715.66 × 10 7
3144120.14341.11 × 10 7 8110.164351.35 × 10 7
3155130.164464.28 × 10 8 8110.158262.14 × 10 7
3169140.0026521.63 × 10 7 680.0056161.91 × 10 8
3178130.002235.69 × 10 7 570.0013734.71 × 10 7
3188130.0020885.13 × 10 7 570.0013194.25 × 10 7
3198130.0020947.71 × 10 7 570.0013296.39 × 10 7
3208130.0023343.29 × 10 7 570.0015552.72 × 10 7
3218130.0021173.45 × 10 7 570.0013482.85 × 10 7
3228130.0020492.32 × 10 7 570.0013931.92 × 10 7
3239140.0054213.64 × 10 7 680.003134.27 × 10 8
3249140.0053921.43 × 10 7 680.0031461.67 × 10 8
3259140.0046361.29 × 10 7 570.0028239.50 × 10 7
3269140.0066211.93 × 10 7 680.0031472.27 × 10 8
3278130.0053177.35 × 10 7 570.0027916.09 × 10 7
3288130.0049597.70 × 10 7 570.0027976.38 × 10 7
3298130.0051015.20 × 10 7 570.0028154.30 × 10 7
3309140.0090095.15 × 10 7 680.0055186.04 × 10 8
3319140.0092072.02 × 10 7 680.0052162.37 × 10 8
3329140.0092981.82 × 10 7 680.006092.13 × 10 8
3339140.0102772.73 × 10 7 680.0054033.21 × 10 8
3349140.0088431.17 × 10 7 570.0067748.61 × 10 7
3359140.0128411.22 × 10 7 570.0057429.03 × 10 7
3368130.0084887.35 × 10 7 570.0055476.09 × 10 7
33710150.0520581.29 × 10 7 680.0241521.35 × 10 7
3389140.0467094.51 × 10 7 680.0275615.29 × 10 8
3399140.0456694.07 × 10 7 680.0227664.77 × 10 8
3409140.0418496.11 × 10 7 680.0273137.17 × 10 8
3419140.0450162.61 × 10 7 680.0247543.06 × 10 8
3429140.0417222.73 × 10 7 680.022723.21 × 10 8
3439140.0405751.84 × 10 7 680.0269912.16 × 10 8
34410150.0983031.83 × 10 7 680.05261.91 × 10 7
3459140.11096.38 × 10 7 680.0485557.49 × 10 8
3469140.0926545.75 × 10 7 680.0498596.75 × 10 8
3479140.0882068.65 × 10 7 680.0483261.01 × 10 7
3489140.0944523.69 × 10 7 680.0499954.32 × 10 8
3499140.0886183.86 × 10 7 680.0564884.53 × 10 8
3509140.0998022.61 × 10 7 680.047953.06 × 10 8

References

  1. Abdullahi, M.; Abubakar, A.B.; Feng, Y.; Liu, J. Comment on: “A derivative-free iterative method for nonlinear monotone equations with convex constraints”. Numer. Algorithms 2023, 94, 1551–1560. [Google Scholar] [CrossRef]
  2. Wang, C.; Wang, Y. A superlinearly convergent projection method for constrained systems of nonlinear equations. J. Glob. Optim. 2009, 44, 283–296. [Google Scholar] [CrossRef]
  3. Ibrahim, A.H.; Kumam, P.; Abubakar, A.B.; Abubakar, J.; Muhammad, A.B. Least-Square-Based Three-Term Conjugate Gradient Projection Method for 1-Norm Problems with Application to Compressed Sensing. Mathematics 2020, 8, 602. [Google Scholar] [CrossRef]
  4. Hager, W.; Zhang, H. A New Conjugate Gradient Method with Guaranteed Descent and an Efficient Line Search. SIAM J. Optim. 2005, 16, 170–192. [Google Scholar] [CrossRef]
  5. Liu, J.K.; Feng, Y. A derivative-free iterative method for nonlinear monotone equations with convex constraints. Numer. Algorithms 2018, 82, 245–262. [Google Scholar] [CrossRef]
  6. Gao, P.; He, C. An efficient three-term conjugate gradient method for nonlinear monotone equations with convex constraints. Calcolo 2018, 55, 53. [Google Scholar] [CrossRef]
  7. Cruz, W.L. A spectral algorithm for large-scale systems of nonlinear monotone equations. Numer. Algorithms 2017, 76, 1109–1130. [Google Scholar] [CrossRef]
  8. Liu, J.; Lu, Z.; Xu, J.; Wu, S.; Tu, Z. An efficient projection-based algorithm without Lipschitz continuity for large-scale nonlinear pseudo-monotone equations. J. Comput. Appl. Math. 2022, 403, 113822. [Google Scholar] [CrossRef]
  9. Awwal, A.M.; Botmart, T. A new sufficiently descent algorithm for pseudomonotone nonlinear operator equations and signal reconstruction. Numer. Algorithms 2023, 94, 1125–1158. [Google Scholar] [CrossRef]
  10. Dolan, E.D.; Moré, J.J. Benchmarking optimization software with performance profiles. Math. Program. 2002, 91, 201–213. [Google Scholar] [CrossRef]
  11. Shehu, Y.; Dong, Q.; Jiang, D. Single projection method for pseudo-monotone variational inequality in Hilbert spaces. Optimization 2019, 68, 385–409. [Google Scholar] [CrossRef]
  12. Thong, D.V.; Vuong, P.T. R-linear convergence analysis of inertial extragradient algorithms for strongly pseudo-monotone variational inequalities. J. Comput. Appl. Math. 2022, 406, 114003. [Google Scholar] [CrossRef]
  13. Jin, X.; Zhang, X.; Huang, K.; Geng, G. Stochastic Conjugate Gradient Algorithm With Variance Reduction. IEEE Trans. Neural Netw. Learn. Syst. 2019, 30, 1360–1369. [Google Scholar] [CrossRef] [PubMed]
  14. Jian, J.; Yin, J.; Tang, C.; Han, D. A family of inertial derivative-free projection methods for constrained nonlinear pseudo-monotone equations with applications. Comput. Appl. Math. 2022, 41, 309. [Google Scholar] [CrossRef]
  15. Liu, W.; Jian, J.; Yin, J. An inertial spectral conjugate gradient projection method for constrained nonlinear pseudo-monotone equations. Numer. Algorithms 2024, 97, 985–1015. [Google Scholar] [CrossRef]
  16. Koorapetse, M.; Kaelo, P.; Lekoko, S.; Diphofu, T. A derivative-free RMIL conjugate gradient projection method for convex constrained nonlinear monotone equations with applications in compressive sensing. Appl. Numer. Math. 2021, 165, 431–441. [Google Scholar] [CrossRef]
  17. Chang, C.; Lin, C. LIBSVM: A library for support vector machines. ACM Trans. Intell. Syst. Technol. (TIST) 2011, 2, 1–27. [Google Scholar] [CrossRef]
  18. La Cruz, W.; Martínez, J.; Raydan, M. Spectral residual method without gradient information for solving large-scale nonlinear systems of equations. Math. Comput. 2006, 75, 1429–1448. [Google Scholar] [CrossRef]
  19. Yu, Z.; Lin, J.; Sun, J.; Xiao, Y.H.; Liu, L.Y.; Li, Z.H. Spectral gradient projection method for monotone nonlinear equations with convex constraints. Appl. Numer. Math. 2009, 59, 2416–2423. [Google Scholar] [CrossRef]
  20. Li, Q.; Li, D. A class of derivative-free methods for large-scale nonlinear monotone equations. IMA J. Numer. Anal. 2011, 31, 1625–1635. [Google Scholar] [CrossRef]
  21. Zhou, W.; Li, D. Limited memory BFGS method for nonlinear monotone equations. J. Comput. Math. 2007, 25, 89–96. [Google Scholar]
  22. Abubakar, A.B.; Kumam, P.; Mohammad, H.; Ibrahim, A.H.; Kiri, A.I. A hybrid approach for finding approximate solutions to constrained nonlinear monotone operator equations with applications. Appl. Numer. Math. 2022, 177, 79–92. [Google Scholar] [CrossRef]
Figure 1. Number of iterations performance profiles.
Figure 1. Number of iterations performance profiles.
Mathematics 12 03734 g001
Figure 2. Number of function evaluations performance profiles.
Figure 2. Number of function evaluations performance profiles.
Mathematics 12 03734 g002
Figure 3. CPU time (in seconds) performance profiles.
Figure 3. CPU time (in seconds) performance profiles.
Mathematics 12 03734 g003
Table 1. Results from the experiments on the pseudo-monotone problems P1 and P2 by Modified Algorithm 2.1 and Algorithm 2.1, respectively.
Table 1. Results from the experiments on the pseudo-monotone problems P1 and P2 by Modified Algorithm 2.1 and Algorithm 2.1, respectively.
Modified Algorithm 2.1Algorithm 2.1
ProblemInitial Point#ITN#FENCPUTIMSNorm#ITN#FENCPUTIMSNorm
P1(0,0)17170.0083974.48 × 10 7 34340.0026637.95 × 10 7
( 1 , 1 ) T 18180.0069647.19 × 10 7 36360.002577.75 × 10 7
( 0.1 , 0.1 ) T 17170.0047367.44 × 10 7 34340.0022939.13 × 10 7
( 1 , 1 ) T 16160.0025529.75 × 10 7 34340.0027866.46 × 10 7
( 0.1 , 0.1 ) T 16160.0045378.75 × 10 7 34340.0025577.00 × 10 7
P2(0.2,0.2,0.2)60660.0179919.63 × 10 7 981000.0092628.98 × 10 7
( 0.4 , 0.4 , 0.4 ) T 61670.003448.96 × 10 7 1021040.0050589.73 × 10 7
( 0.6 , 0.6 , 0.6 ) T 55610.0035599.53 × 10 7 69710.0037759.38 × 10 7
( 0.8 , 0.8 , 0.8 ) T 75810.0041549.09 × 10 7 85870.0044719.81 × 10 7
( 1 , 1 , 1 ) T 78840.0047259.26 × 10 7 56570.00369.75 × 10 7
Table 2. LIBSVM test instances.
Table 2. LIBSVM test instances.
S/NDatasetData Points NNumber of Variables n
1a1a.t30,956123
2a2a.t30,296123
3a3a.t29,376123
4a4a.t27,780123
5a5a.t26,147123
6a6a.t21,341123
7colon-cancer622000
8svmguide1.t40004
9svmguide4.t31210
10usps.t2007256
Table 3. Experimental results on the 10 instances in Table 2.
Table 3. Experimental results on the 10 instances in Table 2.
DFMRMILAlgorithm 2.1Modified Algorithm 2.1
S/N#ITN/#FEN/CPUTIMS/Norm#ITN/#FEN/CPUTIMS/Norm#ITN/#FEN/CPUTIMS/Norm
1715.2/1430.6/7.709/9.92 × 10 7 172.4/345.4/1.881/9.86 × 10 7 116.4/249.6/1.345/9.24 × 10 7
2715.0/1430.0/7.274/9.94 × 10 7 173.0/346.0/1.700/9.70 × 10 7 109.2/237.6/1.182/9.41 × 10 7
3715.2/1430.6/7.181/9.94 × 10 7 172.6/345.8/1.746/9.84 × 10 7 103.8/224.8/1.140/9.18 × 10 7
4714.0/1428.0/6.878/9.95 × 10 7 172.4/345.4/1.669/9.78 × 10 7 110.2/239.2/1.162/8.94 × 10 7
5713.4/1427.0/6.403/9.93 × 10 7 172.2/344.8/1.552/9.86 × 10 7 113.2/245.2/1.107/8.92 × 10 7
6711.8/1423.6/5.078/9.94 × 10 7 171.8/344.2/1.213/9.78 × 10 7 110.6/239.2/0.856/9.23 × 10 7
71257.0/21529.0/5.968/9.68 × 10 7 788.8/3412.6/0.958/9.64 × 10 7 695.6/3191.4/0.900/9.71 × 10 7
8548.6/2329.0/0.164/9.95 × 10 7 156.6/399.0/0.027/9.85 × 10 7 249.0/1020.4/0.070/9.38 × 10 7
9930.4/13502.2/0.158/9.88 × 10 7 652.0/2603.4/0.030/9.71 × 10 7 524.8/2234.4/0.026/9.64 × 10 7
10881.2/9749.2/69.624/9.95 × 10 7 222.2/615.8/4.378/9.68 × 10 7 483.8/2238.0/15.975/9.41 × 10 7
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Share and Cite

MDPI and ACS Style

Muangchoo, K.; Abubakar, A.B. Enhanced Projection Method for the Solution of the System of Nonlinear Equations Under a More General Assumption than Pseudo-Monotonicity and Lipschitz Continuity. Mathematics 2024, 12, 3734. https://doi.org/10.3390/math12233734

AMA Style

Muangchoo K, Abubakar AB. Enhanced Projection Method for the Solution of the System of Nonlinear Equations Under a More General Assumption than Pseudo-Monotonicity and Lipschitz Continuity. Mathematics. 2024; 12(23):3734. https://doi.org/10.3390/math12233734

Chicago/Turabian Style

Muangchoo, Kanikar, and Auwal Bala Abubakar. 2024. "Enhanced Projection Method for the Solution of the System of Nonlinear Equations Under a More General Assumption than Pseudo-Monotonicity and Lipschitz Continuity" Mathematics 12, no. 23: 3734. https://doi.org/10.3390/math12233734

APA Style

Muangchoo, K., & Abubakar, A. B. (2024). Enhanced Projection Method for the Solution of the System of Nonlinear Equations Under a More General Assumption than Pseudo-Monotonicity and Lipschitz Continuity. Mathematics, 12(23), 3734. https://doi.org/10.3390/math12233734

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Metrics

Back to TopTop