An Improved Three-Term Conjugate Gradient Algorithm for Constrained Nonlinear Equations under Non-Lipschitz Conditions and Its Applications
Abstract
:1. Introduction
2. Algorithm
Algorithm 1 Improved Three-Term Conjugate Gradient Algorithm |
Step 0. Choose , , , , and an initial point . Set . Step 1. Set . Step 2. Set the trial point , where the step length satisfies
Step 3. If and , and stop. Otherwise, continue to Step 4. Step 4. Compute the next iterative point as
Step 5. If , stop. Otherwise, compute the search direction by (3). Step 6. Set and go to Step 2. |
3. Convergence Analysis
4. Numerical Experiments
5. Applications in Image Denoising
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
- Garcke, H.; Hüttl, P.; Knopf, P. Shape and topology optimization involving the eigenvalues of an elastic structure: A multi-phase-field approach. Adv. Nonlinear Anal. 2021, 11, 159–197. [Google Scholar] [CrossRef]
- Garcke, H.; Knopf, P.; Yayla, S. Long-time dynamics of the Cahn–Hilliard equation with kinetic rate dependent dynamic boundary conditions. Nonlinear Anal. 2022, 215, 112619. [Google Scholar] [CrossRef]
- Liu, J.; Du, X. A gradient projection method for the sparse signal reconstruction in compressive sensing. Appl. Anal. 2018, 97, 2122–2131. [Google Scholar] [CrossRef]
- Xiao, Y.; Zhu, H. A conjugate gradient method to solve convex constrained monotone equations with applications in compressive sensing. J. Math. Anal. Appl. 2013, 405, 310–319. [Google Scholar] [CrossRef]
- Dirkse, S.; Ferris, M. MCPLIB: A collection of nonlinear mixed complementarity problems. Optim. Methods Softw. 1995, 5, 319–345. [Google Scholar] [CrossRef]
- Wood, A.; Wollenberg, B. Power Generation, Operation, and Control; Wiley: New York, NY, USA, 1996. [Google Scholar]
- Li, D.; Wang, S.; Li, Y.; Wu, J. A convergence analysis of hybrid gradient projection algorithm for constrained nonlinear equations with applications in compressed sensing. Numer. Algorithms 2024, 95, 1325–1345. [Google Scholar] [CrossRef]
- Li, D.; Wu, J.; Li, Y.; Wang, S. A modified spectral gradient projection-based algorithm for large-scale constrained nonlinear equations with applications in compressive sensing. J. Comput. Appl. Math. 2023, 424, 115006. [Google Scholar] [CrossRef]
- Liu, J.; Duan, Y. Two spectral gradient projection methods for constrained equations and their linear convergence rate. J. Inequal. Appl. 2015, 2015, 8. [Google Scholar] [CrossRef]
- Sulaiman, I.M.; Awwal, A.M.; Malik, M. A derivative-free mzprp projection method for convex constrained nonlinear equations and its application in compressive sensing. Mathematics 2022, 10, 2884. [Google Scholar] [CrossRef]
- Li, D.; Wang, S.; Li, Y.; Wu, J. A projection-based hybrid PRP-DY type conjugate gradient algorithm for constrained nonlinear equations with applications. Appl. Numer. Math. 2024, 195, 105–125. [Google Scholar] [CrossRef]
- Yin, J.; Jian, J.; Jiang, X. A generalized hybrid CGPM-based algorithm for solving large-scale convex constrained equations with applications to image restoration. J. Comput. Appl. Math. 2021, 391, 113423. [Google Scholar] [CrossRef]
- Liu, P.; Jian, J.; Jiang, X. A new conjugate gradient projection method for convex constrained nonlinear equations. Complexity 2020, 2020, 8323865. [Google Scholar] [CrossRef]
- Yuan, G.; Li, T.; Hu, W. A conjugate gradient algorithm for large-scale nonlinear equations and image restoration problems. Appl. Numer. Math. 2020, 147, 129–141. [Google Scholar] [CrossRef]
- Ali, E.; Mahdi, S. Adaptive hybrid mixed two-point step size gradient algorithm for solving non-linear systems. Mathematics 2023, 11, 2102. [Google Scholar] [CrossRef]
- Kumam, P.; Abubakar, A.B.; Malik, M. A hybrid HS-LS conjugate gradient algorithm for unconstrained optimization with applications in motion control and image recovery. J. Comput. Appl. Math. 2023, 433, 115304. [Google Scholar] [CrossRef]
- Ullah, N.; Shah, A.; Sabíu, J. A one-parameter memoryless DFP algorithm for solving system of monotone nonlinear equations with application in image processing. Mathematics 2023, 11, 1221. [Google Scholar] [CrossRef]
- Yin, J.; Jian, J.; Jiang, X. A hybrid three-term conjugate gradient projection method for constrained nonlinear monotone equations with applications. Numer. Algorithms 2021, 88, 389–418. [Google Scholar] [CrossRef]
- 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]
- Yuan, G.; Zhang, M. A three-terms Polak–Ribière–Polyak conjugate gradient algorithm for large-scale nonlinear equations. J. Comput. Appl. Math. 2015, 286, 186–195. [Google Scholar] [CrossRef]
- Jiang, X.; Liao, W.; Yin, J. A new family of hybrid three-term conjugate gradient methods with applications in image restoration. Numer. Algorithms 2022, 91, 161–191. [Google Scholar] [CrossRef]
- Liu, Y.; Zhu, Z.; Zhang, B. Two sufficient descent three-term conjugate gradient methods for unconstrained optimization problems with applications in compressive sensing. J. Appl. Math. Comput. 2022, 1–30. [Google Scholar] [CrossRef]
- Kim, H.; Wang, C.; Byun, H. Variable three-term conjugate gradient method for training artificial neural networks. Neural Networks 2023, 159, 125–136. [Google Scholar] [CrossRef]
- Li, M. A modified Hestense-Stiefel conjugate gradient method close to the memoryless BFGS quasi-Newton method. Optim. Methods Softw. 2018, 33, 336–353. [Google Scholar] [CrossRef]
- Li, M. A three-term polak-ribière-polyak conjugate gradient method close to the memoryless BFGS quasi-Newton mthod. J. Ind. Manag. Optim. 2017, 13, 1–16. [Google Scholar]
- Li, M. A family of three-term nonlinear conjugate gradient methods close to the memoryless BFGS method. Optim. Lett. 2018, 12, 1911–1927. [Google Scholar] [CrossRef]
- Ding, Y.; Xiao, Y.; Li, J. A class of conjugate gradient methods for convex constrained monotone equations. Optimization 2017, 66, 2309–2328. [Google Scholar] [CrossRef]
- Zheng, L.; Yang, L.; Liang, Y. A conjugate gradient projection method for solving equations with convex constraints. J. Comput. Appl. Math. 2020, 375, 112781. [Google Scholar] [CrossRef]
- Dolan, E.; Moré, J. Benchmarking optimization software with performance profiles. Math. Program. 2002, 91, 201–213. [Google Scholar] [CrossRef]
- Chan, R.; Ho, C.; Nikolova, M. Salt-and-pepper noise removal by median-type noise detectors and detail-preserving regularization. IEEE Trans. Image Process. 2005, 14, 1479–1485. [Google Scholar] [CrossRef]
- Cai, J.; Chan, R.; Fiore, D. Minimization of a detail-preserving regularization functional for impulse noise removal. J. Math. Imaging Vis. 2007, 29, 79–91. [Google Scholar] [CrossRef]
Init (n) | ITTCG | HTTCGP | ZYL |
---|---|---|---|
Time/Nfunc/Niter/Norm | Time/Nfunc/Niter/Norm | Time/Nfunc/Niter/Norm | |
x1(1) | 8.30/7/1/0.00 | 2.89/7/1/0.00 | 3.77/90/18/9.93 |
x2(1) | 7.84/4/1/0.00 | 1.11/4/1/0.00 | 5.27/25/8/1.84 |
x3(1) | 1.53/78/16/5.89 | 1.56/84/17/4.60 | 4.51/261/62/6.28 |
x4(1) | 1.16/53/11/3.86 | 1.44/69/15/2.46 | 1.84/100/20/5.05 |
x5(1) | 1.93/69/14/6.42 | 8.06/38/8/6.62 | 6.76/288/68/8.66 |
x6(1) | 1.49/53/11/3.94 | 1.52/69/15/3.14 | 2.03/100/20/5.09 |
x7(1) | 1.35/53/11/3.44 | 1.78/69/15/1.28 | 2.96/100/20/5.60 |
x8(1) | 1.28/53/11/8.69 | 2.57/108/21/9.53 | 1.95/100/20/4.68 |
x1(5) | 1.35/7/1/0.00 | 8.77/7/1/0.00 | 1.23/90/18/8.21 |
x2(5) | 5.94/4/1/0.00 | 5.96/4/1/0.00 | 4.38/25/8/1.84 |
x3(5) | 6.18/4/1/0.00 | 4.95/4/1/0.00 | 3.91/25/8/2.00 |
x4(5) | 5.79/44/9/3.64 | 8.06/69/15/2.23 | 1.18/105/21/4.48 |
x5(5) | 8.99/69/14/6.60 | 5.43/38/8/6.39 | 3.46/288/68/8.67 |
x6(5) | 5.24/44/9/3.72 | 8.65/69/15/4.01 | 1.04/105/21/4.49 |
x7(5) | 5.48/44/9/3.98 | 7.85/69/15/3.15 | 1.16/105/21/4.60 |
x8(5) | 6.42/53/11/5.88 | 8.85/69/15/9.92 | 1.16/105/21/4.40 |
x1(10) | 1.28/7/1/0.00 | 1.25/7/1/0.00 | 1.50/90/18/8.80 |
x2(10) | 8.37/4/1/0.00 | 7.26/4/1/0.00 | 5.07/25/8/1.84 |
x3(10) | 7.53/4/1/0.00 | 7.21/4/1/0.00 | 5.80/25/8/2.00 |
x4(10) | 8.52/44/9/1.54 | 1.28/69/15/3.77 | 1.82/105/21/6.34 |
x5(10) | 1.33/69/14/6.62 | 6.92/38/8/6.36 | 5.57/288/68/8.67 |
x6(10) | 9.67/44/9/1.58 | 1.39/69/15/5.06 | 1.91/105/21/6.34 |
x7(10) | 9.01/44/9/1.69 | 1.37/69/15/4.45 | 1.92/105/21/6.42 |
x8(10) | 1.15/53/11/2.05 | 1.39/69/15/2.94 | 1.83/105/21/6.17 |
x1(50) | 5.84/7/1/0.00 | 3.71/7/1/0.00 | 5.84/95/19/5.74 |
x2(50) | 2.75/4/1/0.00 | 2.56/4/1/0.00 | 1.56/25/8/1.84 |
x3(50) | 2.46/4/1/0.00 | 2.31/4/1/0.00 | 1.60/25/8/2.00 |
x4(50) | 5.49/81/17/5.07 | 4.94/69/15/9.57 | 6.51/110/22/5.60 |
x5(50) | 4.53/69/14/6.64 | 2.60/38/8/6.34 | 1.84/288/68/8.67 |
x6(50) | 5.33/76/16/8.44 | 5.10/74/16/6.93 | 6.71/110/22/5.60 |
x7(50) | 5.21/76/16/4.33 | 4.64/69/15/9.88 | 6.74/110/22/5.62 |
x8(50) | 3.59/48/10/0.00 | 5.82/78/17/7.67 | 6.84/110/22/5.55 |
x1(100) | 9.61/7/1/0.00 | 9.45/7/1/0.00 | 1.29/95/19/7.76 |
x2(100) | 6.76/4/1/0.00 | 5.55/4/1/0.00 | 3.12/25/8/1.84 |
x3(100) | 5.52/4/1/0.00 | 5.36/4/1/0.00 | 3.02/25/8/2.00 |
x4(100) | 7.97/52/11/0.00 | 1.12/74/16/6.18 | 1.42/110/22/7.92 |
x5(100) | 9.69/69/14/6.64 | 5.44/38/8/6.33 | 3.61/288/68/8.67 |
x6(100) | 7.86/52/11/0.00 | 1.10/74/16/8.59 | 1.40/110/22/7.92 |
x7(100) | 8.00/52/11/0.00 | 1.15/74/16/7.44 | 1.82/110/22/7.94 |
x8(100) | 1.02/62/13/5.59 | 1.25/78/17/9.11 | 1.43/110/22/7.88 |
Init(n) | ITTCG | HTTCGP | ZYL |
---|---|---|---|
Time/Nfunc/Niter/Norm | Time/Nfunc/Niter/Norm | Time/Nfunc/Niter/Norm | |
x1(1) | 3.05/5/1/0.00 | 1.26/5/1/0.00 | 6.58/36/11/1.82 |
x2(1) | 9.14/4/1/0.00 | 1.24/4/1/0.00 | 4.13/25/8/1.84 |
x3(1) | 4.95/22/8/9.13 | 4.36/23/8/1.61 | 1.46/94/31/5.23 |
x4(1) | 1.06/47/15/2.08 | 1.23/59/19/9.26 | 2.11/117/38/7.52 |
x5(1) | 5.70/24/8/6.49 | 5.69/28/9/6.46 | 1.43/81/26/9.56 |
x6(1) | 9.49/40/13/5.57 | 7.91/38/13/8.06 | 2.14/121/39/4.72 |
x7(1) | 1.06/47/15/2.08 | 1.15/59/19/9.26 | 2.06/117/38/7.52 |
x8(1) | 1.03/39/14/1.88 | 9.06/40/13/4.75 | 1.42/76/24/8.46 |
x1(5) | 5.79/5/1/0.00 | 4.40/5/1/0.00 | 4.36/36/11/4.07 |
x2(5) | 3.91/4/1/0.00 | 4.16/4/1/0.00 | 2.82/25/8/1.84 |
x3(5) | 5.10/4/1/0.00 | 4.63/4/1/0.00 | 3.11/25/8/2.00 |
x4(5) | 6.11/45/15/3.16 | 6.60/55/18/1.08 | 1.27/126/41/8.02 |
x5(5) | 2.69/24/8/9.08 | 2.99/28/9/8.18 | 8.06/81/26/9.57 |
x6(5) | 4.58/44/14/5.94 | 4.47/43/15/6.72 | 1.00/106/34/9.43 |
x7(5) | 4.87/45/15/3.16 | 6.44/55/18/1.08 | 1.11/126/41/8.02 |
x8(5) | 6.10/50/17/7.67 | 4.00/38/13/7.39 | 1.15/123/40/6.51 |
x1(10) | 6.70/5/1/0.00 | 6.54/5/1/0.00 | 4.92/36/11/5.75 |
x2(10) | 5.04/4/1/0.00 | 5.16/4/1/0.00 | 3.03/25/8/1.84 |
x3(10) | 5.41/4/1/0.00 | 6.07/4/1/0.00 | 3.99/25/8/2.00 |
x4(10) | 9.40/52/16/7.66 | 8.07/56/18/5.41 | 1.80/129/42/6.00 |
x5(10) | 4.74/24/8/9.43 | 4.19/28/9/8.42 | 1.18/81/26/9.57 |
x6(10) | 8.57/51/16/8.19 | 8.58/58/19/2.93 | 1.59/112/36/4.86 |
x7(10) | 1.06/52/16/7.66 | 9.62/56/18/5.41 | 1.94/129/42/6.00 |
x8(10) | 1.30/50/17/5.50 | 9.49/58/19/1.09 | 2.12/120/39/8.17 |
x1(50) | 3.13/5/1/0.00 | 1.94/5/1/0.00 | 2.00/39/12/2.06 |
x2(50) | 1.69/4/1/0.00 | 1.39/4/1/0.00 | 1.03/25/8/1.84 |
x3(50) | 1.71/4/1/0.00 | 1.64/4/1/0.00 | 1.12/25/8/2.00 |
x4(50) | 3.51/60/19/4.83 | 3.23/57/18/4.68 | 6.65/129/42/9.70 |
x5(50) | 1.48/24/8/9.72 | 1.68/28/9/8.62 | 4.04/81/26/9.57 |
x6(50) | 3.64/54/18/1.53 | 2.29/39/14/3.40 | 5.72/117/38/5.84 |
x7(50) | 3.77/60/19/4.83 | 3.44/57/18/4.68 | 6.29/129/42/9.70 |
x8(50) | 3.82/58/18/2.83 | 2.89/41/14/4.10 | 6.26/129/42/8.15 |
x1(100) | 4.75/5/1/0.00 | 3.72/5/1/0.00 | 3.23/39/12/2.91 |
x2(100) | 3.49/4/1/0.00 | 2.62/4/1/0.00 | 1.60/25/8/1.84 |
x3(100) | 3.94/4/1/0.00 | 3.03/4/1/0.00 | 1.60/25/8/2.00 |
x4(100) | 4.69/41/15/3.94 | 3.74/41/14/3.81 | 1.06/132/43/8.32 |
x5(100) | 2.63/24/8/9.75 | 2.77/28/9/8.64 | 6.34/81/26/9.57 |
x6(100) | 5.50/47/18/3.93 | 4.14/43/15/4.35 | 1.04/129/42/5.76 |
x7(100) | 4.78/41/15/3.94 | 4.06/41/14/3.81 | 1.10/132/43/8.32 |
x8(100) | 6.30/48/17/3.01 | 5.47/45/16/4.28 | 1.07/129/42/7.87 |
Init(n) | ITTCG | HTTCGP | ZYL |
---|---|---|---|
Time/Nfunc/Niter/Norm | Time/Nfunc/Niter/Norm | Time/Nfunc/Niter/Norm | |
x1(1) | 5.36/250/33/5.87 | 5.72/293/36/9.74 | 8.26/439/73/9.64 |
x2(1) | 3.93/180/23/7.99 | 5.81/307/38/8.37 | 7.07/367/61/9.94 |
x3(1) | 4.19/208/27/8.37 | 6.63/349/43/8.55 | 7.49/385/64/8.74 |
x4(1) | 4.51/222/29/6.50 | 8.57/433/54/9.46 | 7.66/391/65/9.70 |
x5(1) | 4.73/237/31/6.04 | 6.64/337/42/8.37 | 7.42/391/65/9.80 |
x6(1) | 4.61/222/29/6.49 | 8.58/410/51/9.15 | 8.01/397/66/9.25 |
x7(1) | 5.08/250/33/5.41 | 6.35/330/41/8.54 | 7.97/421/70/9.17 |
x8(1) | 6.45/308/41/5.09 | 8.47/413/51/9.89 | 9.79/480/80/9.21 |
x1(5) | 3.76/243/32/6.03 | 3.64/299/37/8.62 | 5.50/445/74/8.19 |
x2(5) | 2.20/173/22/8.68 | 3.98/338/42/1.00 | 4.94/403/67/9.85 |
x3(5) | 2.43/201/26/8.32 | 3.75/319/39/9.75 | 4.98/415/69/9.21 |
x4(5) | 2.67/229/30/8.15 | 4.77/418/52/3.03 | 5.21/421/70/7.23 |
x5(5) | 2.62/216/28/5.92 | 3.86/316/39/7.17 | 5.17/415/69/8.21 |
x6(5) | 2.92/229/30/8.14 | 5.12/417/52/9.34 | 5.52/415/69/8.50 |
x7(5) | 3.48/271/36/7.63 | 7.09/560/70/9.12 | 5.42/439/73/7.95 |
x8(5) | 4.21/309/41/5.34 | 5.09/413/51/5.52 | 6.69/522/87/9.37 |
x1(10) | 4.56/236/31/6.73 | 6.37/338/42/3.92 | 8.78/451/75/9.13 |
x2(10) | 3.39/187/24/9.18 | 6.63/338/42/9.20 | 7.80/415/69/9.61 |
x3(10) | 3.83/208/27/9.16 | 6.19/335/42/6.78 | 7.91/427/71/9.89 |
x4(10) | 4.54/236/31/7.88 | 6.21/359/45/4.72 | 6.89/397/66/6.98 |
x5(10) | 4.12/223/29/5.94 | 5.37/302/37/5.30 | 7.13/409/68/9.04 |
x6(10) | 4.33/236/31/7.88 | 6.24/369/46/3.53 | 6.95/397/66/7.00 |
x7(10) | 5.05/264/35/8.21 | 9.77/568/71/5.46 | 7.02/397/66/9.17 |
x8(10) | 5.65/309/41/7.30 | 6.72/387/48/8.03 | 8.77/486/81/7.25 |
x1(50) | 1.75/236/31/7.21 | 2.01/292/36/6.60 | 3.00/427/71/9.67 |
x2(50) | 1.31/180/23/9.41 | 2.50/350/44/9.18 | 2.97/427/71/9.94 |
x3(50) | 1.44/209/27/4.76 | 2.80/412/52/8.85 | 3.02/415/69/7.96 |
x4(50) | 1.60/229/30/8.95 | 2.65/394/50/8.74 | 2.90/416/69/9.13 |
x5(50) | 1.62/237/31/6.63 | 1.95/288/36/9.73 | 3.12/439/73/8.44 |
x6(50) | 1.52/229/30/8.95 | 2.64/387/49/8.40 | 2.90/416/69/9.17 |
x7(50) | 1.80/264/35/8.82 | 2.63/381/48/7.31 | 2.79/410/68/9.11 |
x8(50) | 2.24/324/43/6.03 | 3.16/451/56/7.36 | 3.41/481/80/9.57 |
x1(100) | 4.19/222/29/8.67 | 5.07/261/32/7.78 | 8.75/433/72/8.25 |
x2(100) | 3.24/180/23/9.00 | 5.38/265/33/7.48 | 8.16/415/69/8.73 |
x3(100) | 3.84/216/28/4.70 | 6.33/320/40/4.66 | 7.70/415/69/9.78 |
x4(100) | 4.77/243/32/9.84 | 5.49/378/48/8.73 | 7.55/416/69/7.31 |
x5(100) | 4.32/230/30/7.85 | 6.88/344/43/9.43 | 9.14/451/75/9.94 |
x6(100) | 4.56/243/32/9.84 | 6.98/347/44/9.61 | 8.11/416/69/7.31 |
x7(100) | 4.67/244/32/5.07 | 6.55/317/40/6.90 | 8.56/422/70/9.36 |
x8(100) | 6.35/345/46/8.22 | 9.66/475/59/7.73 | 9.91/493/82/8.45 |
Init(n) | ITTCG | HTTCGP | ZYL |
---|---|---|---|
Time/Nfunc/Niter/Norm | Time/Nfunc/Niter/Norm | Time/Nfunc/Niter/Norm | |
x1(1) | 2.21/89/20/2.89 | 2.83/119/27/8.70 | 2.78/118/38/8.33 |
x2(1) | 1.57/51/19/1.61 | 1.56/56/21/9.69 | 2.28/96/33/6.03 |
x3(1) | 1.23/39/15/8.04 | 1.53/54/18/3.13 | 2.20/93/32/7.55 |
x4(1) | 4.91/223/39/4.76 | 4.59/237/35/7.19 | 2.80/123/39/8.83 |
x5(1) | 9.47/29/12/4.29 | 1.40/52/19/6.57 | 2.37/102/35/8.01 |
x6(1) | 4.47/212/36/8.45 | 5.37/279/43/1.69 | 2.77/123/39/8.74 |
x7(1) | 1.12/37/14/1.06 | 1.41/53/19/8.14 | 2.05/87/30/7.34 |
x8(1) | 6.41/328/44/6.49 | 6.04/320/44/7.16 | 4.16/207/50/6.30 |
x1(5) | 2.23/231/36/1.93 | 1.77/216/33/5.67 | 1.50/132/42/7.97 |
x2(5) | 5.03/39/16/7.13 | 6.34/50/18/4.49 | 1.02/102/35/6.85 |
x3(5) | 5.63/39/16/8.49 | 7.93/68/26/5.31 | 1.03/96/33/6.96 |
x4(5) | 3.01/373/49/6.04 | 4.30/567/65/2.84 | 1.33/133/42/8.46 |
x5(5) | 4.56/36/14/2.40 | 7.41/62/23/9.87 | 9.95/99/34/6.30 |
x6(5) | 3.07/376/50/1.06 | 4.29/570/62/1.23 | 1.31/141/45/7.46 |
x7(5) | 6.42/45/17/3.98 | 6.82/61/22/7.17 | 1.05/96/33/5.87 |
x8(5) | 4.89/637/69/9.82 | 4.17/562/63/6.41 | 1.70/189/44/9.43 |
x1(10) | 2.08/93/24/3.59 | 3.40/265/34/3.37 | 2.11/118/38/7.24 |
x2(10) | 1.13/42/16/1.55 | 6.60/35/13/3.54 | 2.09/108/37/5.11 |
x3(10) | 1.12/45/17/3.65 | 1.33/55/20/9.85 | 1.98/108/37/5.70 |
x4(10) | 6.45/464/57/8.15 | 1.00/811/87/5.54 | 2.51/138/44/7.43 |
x5(10) | 1.15/37/15/9.00 | 9.85/51/19/2.68 | 1.91/102/35/5.79 |
x6(10) | 6.39/461/56/9.13 | 1.17/1002/101/2.35 | 2.36/138/44/9.78 |
x7(10) | 1.23/43/17/4.03 | 1.32/53/21/7.96 | 2.08/109/37/8.60 |
x8(10) | 8.19/633/66/5.97 | 1.11/1005/92/2.54 | 4.14/276/54/8.07 |
x1(50) | 1.73/401/46/1.44 | 2.90/724/73/8.61 | 9.25/186/46/7.62 |
x2(50) | 4.29/57/20/2.84 | 4.22/65/22/4.43 | 6.37/108/37/8.56 |
x3(50) | 3.75/57/19/3.18 | 3.22/46/18/8.01 | 6.17/108/37/8.41 |
x4(50) | 3.34/788/76/5.24 | 8.21/2118/166/6.17 | 8.29/139/43/9.48 |
x5(50) | 3.79/45/17/2.58 | 5.17/70/24/4.35 | 6.62/108/37/8.69 |
x6(50) | 3.40/805/78/9.22 | 6.01/1542/130/6.18 | 7.45/133/41/7.02 |
x7(50) | 4.43/58/21/1.15 | 4.87/75/26/3.42 | 6.55/112/38/9.59 |
x8(50) | 4.03/968/91/8.15 | 7.75/2002/154/3.01 | 1.63/371/61/9.71 |
x1(100) | 3.19/422/51/5.34 | 6.89/943/93/3.29 | 2.17/267/58/9.06 |
x2(100) | 5.56/42/16/6.16 | 8.42/75/25/3.72 | 1.19/118/40/4.98 |
x3(100) | 5.52/42/16/6.54 | 1.17/103/33/7.65 | 1.17/118/40/5.08 |
x4(100) | 5.75/771/77/2.88 | 1.34/1932/156/7.33 | 1.24/133/39/7.09 |
x5(100) | 8.06/66/23/7.79 | 7.31/58/20/1.15 | 1.17/118/40/5.29 |
x6(100) | 5.64/774/80/6.88 | 1.28/1883/150/7.31 | 1.25/133/39/7.55 |
x7(100) | 8.95/76/24/2.56 | 8.48/71/25/6.40 | 1.10/109/37/9.37 |
x8(100) | 8.61/1225/105/7.94 | 1.92/2868/204/7.36 | 2.37/320/49/9.81 |
Init(n) | ITTCG | HTTCGP | ZYL |
---|---|---|---|
Time/Nfunc/Niter/Norm | Time/Nfunc/Niter/Norm | Time/Nfunc/Niter/Norm | |
x1(1) | 3.96/12/5/5.48 | 5.80/24/10/3.72 | 5.63/31/10/6.08 |
x2(1) | 2.51/10/4/1.65 | 3.69/18/7/5.57 | 3.90/25/8/3.43 |
x3(1) | 5.97/20/7/8.62 | 8.13/33/11/8.36 | 1.04/52/17/5.45 |
x4(1) | 1.06/42/16/6.01 | 8.85/44/14/1.09 | 1.57/88/29/8.47 |
x5(1) | 6.14/26/9/6.17 | 1.25/61/19/6.07 | 1.39/79/26/9.94 |
x6(1) | 1.04/40/16/4.36 | 7.10/35/11/4.13 | 1.69/88/29/9.04 |
x7(1) | 1.08/42/16/6.01 | 9.01/44/14/1.09 | 1.56/88/29/8.47 |
x8(1) | 7.76/32/11/5.11 | 1.13/56/17/3.91 | 1.45/79/26/7.43 |
x1(5) | 2.17/12/5/1.39 | 3.04/24/10/8.33 | 3.11/34/11/2.17 |
x2(5) | 1.18/10/4/1.65 | 1.87/18/7/5.57 | 2.36/25/8/3.43 |
x3(5) | 9.94/8/3/2.00 | 1.70/14/5/7.19 | 2.78/25/8/4.88 |
x4(5) | 8.32/37/13/5.39 | 5.83/47/16/5.29 | 9.94/91/30/9.14 |
x5(5) | 3.12/26/9/6.15 | 3.98/39/14/5.16 | 7.61/79/26/9.95 |
x6(5) | 4.18/37/13/5.25 | 5.90/56/18/6.52 | 8.74/91/30/9.26 |
x7(5) | 4.40/37/13/5.39 | 4.87/47/16/5.29 | 1.31/91/30/9.14 |
x8(5) | 5.04/37/13/5.20 | 4.02/40/13/3.87 | 8.56/91/30/7.14 |
x1(10) | 2.64/12/5/2.15 | 5.83/27/11/4.48 | 4.55/34/11/3.08 |
x2(10) | 1.68/10/4/1.65 | 2.94/18/7/5.57 | 3.62/25/8/3.43 |
x3(10) | 1.27/8/3/2.00 | 2.51/14/5/7.19 | 3.17/25/8/4.88 |
x4(10) | 7.97/40/14/2.70 | 8.08/49/17/7.74 | 1.51/97/32/4.85 |
x5(10) | 4.56/26/9/6.15 | 9.67/42/16/1.01 | 1.26/79/26/9.95 |
x6(10) | 1.00/40/14/2.69 | 7.89/45/16/4.43 | 1.55/97/32/4.88 |
x7(10) | 8.85/40/14/2.70 | 9.63/49/17/7.74 | 1.61/97/32/4.85 |
x8(10) | 7.50/37/13/9.78 | 1.09/59/20/4.61 | 1.50/91/30/9.04 |
x1(50) | 8.56/12/5/6.49 | 2.03/27/11/1.00 | 1.79/34/11/6.88 |
x2(50) | 5.63/10/4/1.65 | 9.17/18/7/5.57 | 9.69/25/8/3.43 |
x3(50) | 4.62/8/3/2.00 | 6.75/14/5/7.19 | 9.82/25/8/4.88 |
x4(50) | 2.99/40/14/3.65 | 3.13/47/17/4.20 | 5.09/100/33/8.67 |
x5(50) | 1.85/26/9/6.15 | 2.43/39/13/7.07 | 4.00/79/26/9.95 |
x6(50) | 3.18/40/14/3.65 | 2.60/42/14/8.30 | 5.30/100/33/8.68 |
x7(50) | 2.55/40/14/3.65 | 3.04/47/17/4.20 | 5.10/100/33/8.67 |
x8(50) | 2.91/40/14/3.61 | 3.75/63/20/7.11 | 4.94/97/32/9.80 |
x1(100) | 1.97/14/6/1.09 | 3.18/27/11/1.42 | 3.09/34/11/9.72 |
x2(100) | 9.06/10/4/1.65 | 1.52/18/7/5.57 | 1.74/25/8/3.43 |
x3(100) | 7.64/8/3/2.00 | 1.07/14/5/7.19 | 1.55/25/8/4.88 |
x4(100) | 4.34/40/14/7.05 | 4.65/45/16/3.74 | 8.37/103/34/5.77 |
x5(100) | 2.98/26/9/6.15 | 4.12/41/14/4.50 | 6.93/79/26/9.95 |
x6(100) | 4.34/40/14/7.05 | 4.32/41/14/6.54 | 8.85/103/34/5.78 |
x7(100) | 5.00/40/14/7.05 | 4.74/45/16/3.74 | 9.16/103/34/5.77 |
x8(100) | 4.77/40/14/7.04 | 6.10/54/19/6.43 | 8.39/103/34/5.47 |
Init(n) | ITTCG | HTTCGP | ZYL |
---|---|---|---|
Time/Nfunc/Niter/Norm | Time/Nfunc/Niter/Norm | Time/Nfunc/Niter/Norm | |
x1(1) | 2.71/6/1/0.00 | 2.30/6/1/0.00 | 1.25/57/9/6.00 |
x2(1) | 1.63/9/1/0.00 | 1.48/9/1/0.00 | 6.92/43/7/1.85 |
x3(1) | 1.63/96/12/3.75 | 2.41/151/19/2.45 | 2.70/174/28/9.86 |
x4(1) | 2.26/114/14/5.42 | 3.11/163/20/0.00 | 4.31/219/35/5.14 |
x5(1) | 1.66/81/11/7.63 | 2.88/152/19/3.07 | 3.36/182/29/4.94 |
x6(1) | 2.33/115/15/0.00 | 3.59/200/24/0.00 | 4.57/237/38/2.96 |
x7(1) | 2.25/114/14/5.42 | 3.02/163/20/0.00 | 4.16/219/35/5.14 |
x8(1) | 2.19/95/12/1.03 | 3.90/185/23/2.29 | 4.09/186/30/5.18 |
x1(5) | 1.31/6/1/0.00 | 1.12/6/1/0.00 | 7.51/63/10/1.25 |
x2(5) | 1.05/9/1/0.00 | 8.05/9/1/0.00 | 4.20/43/7/1.85 |
x3(5) | 3.70/3/1/0.00 | 4.07/3/1/0.00 | 3.86/3/1/0.00 |
x4(5) | 1.65/179/22/3.86 | 1.36/163/21/9.04 | 2.00/244/39/4.27 |
x5(5) | 1.26/136/18/0.00 | 1.13/143/18/2.74 | 1.59/194/31/8.94 |
x6(5) | 1.58/185/23/4.39 | 1.13/141/18/2.33 | 2.13/255/41/4.07 |
x7(5) | 1.55/179/22/3.86 | 1.34/163/21/9.04 | 1.94/244/39/4.27 |
x8(5) | 7.83/81/10/0.00 | 1.40/154/20/8.32 | 1.93/225/36/4.26 |
x1(10) | 9.42/6/1/0.00 | 8.92/6/1/0.00 | 6.75/63/10/1.76 |
x2(10) | 8.31/9/1/0.00 | 8.03/9/1/0.00 | 4.85/43/7/1.85 |
x3(10) | 5.54/3/1/0.00 | 4.58/3/1/0.00 | 3.31/3/1/0.00 |
x4(10) | 1.20/105/13/1.67 | 1.65/157/20/5.74 | 2.66/244/39/4.95 |
x5(10) | 1.61/122/16/0.00 | 1.56/154/19/0.00 | 1.93/182/29/4.08 |
x6(10) | 1.10/90/11/2.02 | 1.44/134/17/2.39 | 2.34/226/36/7.42 |
x7(10) | 1.21/105/13/1.67 | 1.60/157/20/5.74 | 2.61/244/39/4.95 |
x8(10) | 1.21/98/12/0.00 | 1.66/131/17/0.00 | 2.84/255/41/8.67 |
x1(50) | 3.23/6/1/0.00 | 2.94/6/1/0.00 | 2.81/63/10/3.94 |
x2(50) | 2.65/9/1/0.00 | 2.75/9/1/0.00 | 1.46/43/7/1.85 |
x3(50) | 1.44/3/1/0.00 | 1.66/3/1/0.00 | 1.39/3/1/0.00 |
x4(50) | 4.61/103/13/1.41 | 6.50/157/20/3.17 | 1.04/250/40/5.94 |
x5(50) | 8.27/186/24/9.60 | 6.28/148/19/3.83 | 7.85/194/31/5.49 |
x6(50) | 4.52/103/13/1.43 | 5.55/134/17/2.22 | 1.00/238/38/5.59 |
x7(50) | 4.50/103/13/1.41 | 6.59/157/20/3.17 | 1.06/250/40/5.94 |
x8(50) | 4.76/96/12/1.33 | 5.49/117/15/8.48 | 1.03/238/38/6.93 |
x1(100) | 8.14/6/1/0.00 | 6.15/6/1/0.00 | 4.80/63/10/5.57 |
x2(100) | 5.83/9/1/0.00 | 4.69/9/1/0.00 | 2.41/43/7/1.85 |
x3(100) | 3.14/3/1/0.00 | 2.90/3/1/0.00 | 2.63/3/1/0.00 |
x4(100) | 1.27/166/21/2.91 | 1.31/173/22/5.65 | 1.81/250/40/9.26 |
x5(100) | 1.04/146/19/0.00 | 1.04/148/19/5.90 | 1.18/175/28/1.08 |
x6(100) | 1.25/166/21/3.23 | 9.50/134/17/2.71 | 1.71/238/38/9.99 |
x7(100) | 1.19/166/21/2.91 | 1.23/173/22/5.65 | 1.76/250/40/9.26 |
x8(100) | 9.91/111/14/9.76 | 1.06/126/16/3.64 | 2.06/273/44/9.01 |
Init(n) | ITTCG | HTTCGP | ZYL |
---|---|---|---|
Time/Nfunc/Niter/Norm | Time/Nfunc/Niter/Norm | Time/Nfunc/Niter/Norm | |
x1(1) | 9.76/90/19/1.33 | 1.35/120/24/3.94 | 3.80/381/90/9.43 |
x2(1) | 8.69/84/18/8.04 | 1.28/133/27/6.50 | 4.29/434/103/6.56 |
x3(1) | 1.10/98/21/6.30 | 1.32/132/26/9.64 | 4.17/426/101/6.50 |
x4(1) | 9.65/102/22/8.39 | 1.07/109/22/5.40 | 4.01/418/99/5.99 |
x5(1) | 7.44/76/16/9.56 | 9.36/96/19/6.21 | 4.00/401/95/7.48 |
x6(1) | 1.01/101/22/9.38 | 1.17/117/23/2.55 | 4.17/418/99/5.98 |
x7(1) | 8.83/89/19/3.19 | 1.24/129/25/2.52 | 3.56/357/84/5.14 |
x8(1) | 8.88/91/20/2.65 | 1.71/167/33/4.74 | 4.15/417/99/9.79 |
x1(5) | 4.95/94/20/5.32 | 5.72/111/22/5.98 | 1.94/369/87/5.58 |
x2(5) | 5.82/113/25/1.86 | 6.32/129/27/5.34 | 2.32/458/109/6.29 |
x3(5) | 5.97/112/25/8.38 | 6.88/137/29/4.35 | 2.40/458/109/6.29 |
x4(5) | 5.91/111/24/2.77 | 4.96/94/19/7.52 | 2.31/446/106/5.82 |
x5(5) | 5.37/103/23/1.59 | 8.01/166/31/5.77 | 2.40/458/109/6.07 |
x6(5) | 6.43/122/27/6.59 | 4.75/94/19/6.85 | 2.28/446/106/5.81 |
x7(5) | 5.11/96/21/3.29 | 5.95/115/23/8.71 | 2.29/454/108/6.09 |
x8(5) | 5.29/103/22/5.52 | 5.48/111/23/7.31 | 2.06/405/96/7.94 |
x1(10) | 1.14/118/26/7.92 | 1.24/132/27/4.22 | 4.45/474/113/7.04 |
x2(10) | 1.04/108/24/5.78 | 1.06/112/23/4.30 | 3.78/395/94/6.07 |
x3(10) | 1.04/109/24/5.61 | 9.55/101/21/4.62 | 3.72/395/94/6.07 |
x4(10) | 1.15/123/27/4.37 | 1.13/122/25/6.26 | 4.19/446/106/7.30 |
x5(10) | 1.29/135/30/4.75 | 1.22/132/27/1.79 | 3.48/370/88/9.63 |
x6(10) | 1.24/130/29/5.79 | 1.19/129/27/7.05 | 4.25/446/106/7.30 |
x7(10) | 1.07/112/25/2.50 | 1.20/127/27/4.10 | 4.28/444/106/9.47 |
x8(10) | 1.05/109/24/3.00 | 1.24/133/27/9.07 | 4.46/466/111/6.03 |
x1(50) | 6.01/118/26/8.42 | 5.36/106/22/8.97 | 2.45/468/112/7.75 |
x2(50) | 5.22/102/23/3.61 | 6.67/130/27/2.68 | 2.42/464/111/8.61 |
x3(50) | 5.24/102/23/3.92 | 6.79/133/27/9.03 | 2.39/464/111/8.57 |
x4(50) | 5.01/98/22/6.48 | 7.64/150/31/2.72 | 2.35/452/108/8.91 |
x5(50) | 5.68/111/25/4.20 | 7.90/155/32/7.32 | 2.41/464/111/8.34 |
x6(50) | 5.04/98/22/6.48 | 7.17/141/29/5.02 | 2.34/452/108/8.91 |
x7(50) | 6.01/115/26/4.87 | 6.63/128/27/9.58 | 2.40/460/110/9.28 |
x8(50) | 5.11/98/22/7.88 | 6.79/131/28/5.48 | 2.30/432/103/7.76 |
x1(100) | 9.81/97/22/8.53 | 1.08/112/23/5.11 | 4.53/456/109/9.55 |
x2(100) | 1.19/118/27/8.34 | 1.33/129/26/4.22 | 4.91/497/119/6.39 |
x3(100) | 1.25/127/29/2.67 | 1.38/143/30/9.45 | 4.77/481/115/6.26 |
x4(100) | 1.06/107/24/3.28 | 1.17/122/25/7.53 | 4.52/456/109/8.83 |
x5(100) | 9.93/101/23/6.39 | 1.14/119/24/4.93 | 4.92/493/118/6.21 |
x6(100) | 1.06/107/24/3.29 | 1.25/128/26/3.82 | 4.56/456/109/8.83 |
x7(100) | 9.94/98/22/9.49 | 1.25/128/26/3.81 | 4.85/485/116/6.04 |
x8(100) | 1.02/102/23/3.43 | 1.09/113/23/5.86 | 4.82/485/116/6.34 |
Init(n) | ITTCG | HTTCGP | ZYL |
---|---|---|---|
Time/Nfunc/Niter/Norm | Time/Nfunc/Niter/Norm | Time/Nfunc/Niter/Norm | |
x1(1) | 4.04/44/14/6.03 | 1.97/26/8/4.64 | 2.25/29/7/5.31 |
x2(1) | 1.03/159/23/6.90 | 1.31/205/28/9.90 | 1.02/160/28/5.49 |
x3(1) | 9.97/164/24/8.26 | 1.26/215/30/7.18 | 9.64/159/28/5.57 |
x4(1) | 7.67/128/19/9.28 | 8.69/140/20/5.23 | 9.64/159/30/4.97 |
x5(1) | 9.63/159/23/7.20 | 1.32/213/30/6.66 | 9.20/155/27/4.53 |
x6(1) | 7.92/128/19/9.28 | 8.39/144/20/5.73 | 1.03/159/30/4.88 |
x7(1) | 7.57/128/19/7.06 | 9.54/155/22/5.75 | 9.33/155/29/9.48 |
x8(1) | 1.15/187/27/5.73 | 1.32/221/30/5.32 | 7.11/120/20/9.25 |
x1(5) | 2.10/47/15/3.88 | 1.00/26/8/1.04 | 1.21/33/8/6.70 |
x2(5) | 5.09/165/24/7.30 | 6.85/230/32/9.16 | 4.22/144/25/8.58 |
x3(5) | 5.39/177/26/9.37 | 6.29/213/30/9.98 | 4.46/148/26/4.02 |
x4(5) | 4.28/133/20/9.06 | 4.55/147/21/7.30 | 4.27/137/26/4.54 |
x5(5) | 5.68/183/27/7.21 | 6.18/197/27/5.19 | 4.93/155/27/7.52 |
x6(5) | 4.07/133/20/9.04 | 5.31/175/25/7.40 | 4.29/137/26/4.73 |
x7(5) | 4.01/133/20/5.99 | 5.52/182/26/5.60 | 4.15/135/26/9.26 |
x8(5) | 6.15/206/30/5.21 | 7.30/239/33/2.79 | 4.02/126/21/5.03 |
x1(10) | 3.57/47/15/5.49 | 1.98/26/8/1.47 | 2.13/33/8/9.47 |
x2(10) | 9.53/165/24/8.88 | 1.21/215/30/5.80 | 8.14/144/25/6.81 |
x3(10) | 9.90/172/25/5.14 | 1.15/202/28/3.43 | 8.20/136/24/8.97 |
x4(10) | 7.54/133/20/4.98 | 7.96/144/21/3.17 | 8.38/141/27/4.63 |
x5(10) | 9.17/165/24/9.01 | 1.07/193/27/7.56 | 8.60/150/26/5.92 |
x6(10) | 7.73/133/20/4.98 | 1.12/206/29/3.42 | 8.49/141/27/4.68 |
x7(10) | 7.29/126/19/8.04 | 1.17/204/29/4.02 | 8.39/140/27/8.91 |
x8(10) | 1.13/200/29/7.31 | 1.32/222/30/3.73 | 7.25/126/21/6.66 |
x1(50) | 1.83/50/16/3.53 | 9.11/26/8/3.28 | 1.12/33/8/2.12 |
x2(50) | 5.21/171/25/8.48 | 5.41/177/25/7.99 | 4.51/143/25/5.60 |
x3(50) | 5.16/171/25/8.84 | 7.61/251/35/5.12 | 4.13/131/23/8.83 |
x4(50) | 4.21/138/21/5.04 | 4.63/150/22/8.39 | 4.68/150/29/9.54 |
x5(50) | 5.25/171/25/8.61 | 7.06/231/33/7.37 | 4.45/143/25/4.76 |
x6(50) | 4.20/138/21/5.04 | 4.90/164/24/3.00 | 4.77/150/29/9.67 |
x7(50) | 4.06/131/20/6.58 | 4.65/152/22/7.21 | 5.08/156/30/4.61 |
x8(50) | 5.92/195/28/8.22 | 8.47/281/39/2.74 | 4.11/132/22/6.17 |
x1(100) | 3.58/50/16/4.99 | 1.80/26/8/4.64 | 2.12/33/8/3.00 |
x2(100) | 1.05/177/26/5.90 | 1.42/244/34/4.53 | 8.12/137/24/6.64 |
x3(100) | 1.04/178/26/5.25 | 9.33/158/22/9.92 | 7.95/131/23/7.50 |
x4(100) | 7.89/131/20/5.53 | 1.33/226/33/7.44 | 9.26/149/29/6.70 |
x5(100) | 1.06/177/26/5.21 | 1.47/242/34/5.15 | 8.52/137/24/7.67 |
x6(100) | 7.95/131/20/5.53 | 1.26/206/30/2.16 | 9.38/149/29/6.70 |
x7(100) | 7.34/124/19/8.94 | 8.96/151/22/7.80 | 9.80/158/31/6.50 |
x8(100) | 1.20/202/29/5.27 | 1.35/231/32/9.96 | 7.90/132/22/9.92 |
Image | ITTCG | HTTCGP | ZYL |
---|---|---|---|
/// | /// | /// | |
lighthouse | 28.2/5.73/30.85/0.97 | 50.5/11.51/31.32/0.97 | 58.7/14.24/31.12/0.97 |
peppers | 19.4/1.18/33.24/0.96 | 51.5/3.19/33.77/0.96 | 29.4/1.94/33.37/0.96 |
boat | 14.4/3.19/33.96/0.98 | 48.0/11.00/34.46/0.98 | 24.4/5.98/34.08/0.98 |
kiel | 25.5/5.25/27.81/0.97 | 49.0/10.93/27.94/0.97 | 47.5/11.37/27.88/0.97 |
fruits | 21.3/1.33/30.01/0.94 | 69.3/4.34/30.25/0.95 | 27.1/1.80/30.01/0.94 |
brain | 14.2/0.90/31.07/0.87 | 23.9/1.51/31.13/0.87 | 18.7/1.28/31.10/0.87 |
clown | 14.1/3.15/36.62/0.99 | 42.4/9.68/37.17/0.99 | 24.9/6.02/36.80/0.99 |
couple | 14.4/3.15/34.26/0.99 | 36.1/8.38/34.60/0.99 | 24.2/5.90/34.33/0.99 |
trucks | 12.1/2.77/34.07/0.98 | 25.0/5.89/34.13/0.98 | 18.4/4.66/34.08/0.98 |
baboon | 32.9/1.72/24.68/0.87 | 35.8/1.85/24.67/0.87 | 72.1/4.47/24.68/0.87 |
barbara | 15.6/3.31/29.03/0.96 | 46.1/10.39/29.07/0.96 | 27.2/6.47/29.04/0.96 |
Cameraman | 19.0/1.13/29.95/0.96 | 37.6/2.33/30.14/0.96 | 38.9/2.54/30.13/0.96 |
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. |
© 2024 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 (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Li, D.; Li, Y.; Wang, S. An Improved Three-Term Conjugate Gradient Algorithm for Constrained Nonlinear Equations under Non-Lipschitz Conditions and Its Applications. Mathematics 2024, 12, 2556. https://doi.org/10.3390/math12162556
Li D, Li Y, Wang S. An Improved Three-Term Conjugate Gradient Algorithm for Constrained Nonlinear Equations under Non-Lipschitz Conditions and Its Applications. Mathematics. 2024; 12(16):2556. https://doi.org/10.3390/math12162556
Chicago/Turabian StyleLi, Dandan, Yong Li, and Songhua Wang. 2024. "An Improved Three-Term Conjugate Gradient Algorithm for Constrained Nonlinear Equations under Non-Lipschitz Conditions and Its Applications" Mathematics 12, no. 16: 2556. https://doi.org/10.3390/math12162556