
-
Previous Article
Maxillofacial surgical simulation system with haptic feedback
- JIMO Home
- This Issue
-
Next Article
Can the reform of green credit policy promote enterprise eco-innovation? A theoretical analysis
A proximal ADMM with the Broyden family for convex optimization problems
Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan |
Alternating direction methods of multipliers (ADMM) have been well studied and effectively used in various application fields. The classical ADMM must solve two subproblems exactly at each iteration. To overcome the difficulty of computing the exact solution of the subproblems, some proximal terms are added to the subproblems. Recently, {{a special proximal ADMM has been studied}} whose regularized matrix in the proximal term is generated by the BFGS update (or limited memory BFGS) at every iteration for a structured quadratic optimization problem. {{The numerical experiments also showed}} that the numbers of iterations were almost same as those by the exact ADMM. In this paper, we propose such a proximal ADMM for more general convex optimization problems, and extend the proximal term by the Broyden family update. We also show the convergence of the proposed method under standard assumptions.
References:
[1] |
S. Banert, R. I. Bot and E. R. Csetnek, Fixing and extending some recent results on the ADMM algorithm, preprint, arXiv: 1612.05057. Google Scholar |
[2] |
S. Boyd, N. Parikh, E. Chu, B. Peleato and J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers, Foundations and TrendsⓇ in Machine Learning, 3 (2011), 1–122. Google Scholar |
[3] |
G. Chen and M. Teboulle,
A proximal-based decomposition method for convex minimization problems, Math. Programming, 64 (1994), 81-101.
doi: 10.1007/BF01582566. |
[4] |
W. Deng and W. Yin,
On the global and linear convergence of the generalized alternating direction method of multipliers, J. Sci. Comput., 66 (2016), 889-916.
doi: 10.1007/s10915-015-0048-x. |
[5] |
J. Eckstein and D. P. Bertsekas,
On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators, Math. Programming, 55 (1992), 293-318.
doi: 10.1007/BF01581204. |
[6] |
J. Eckstein and M. Fukushima, Some reformulations and applications of the alternating direction method of multipliers, in Large Scale Optimization (Gainesville, FL, 1993), Kluwer Acad. Publ., Dordrecht, 1994,115–134.
doi: 10.1007/978-1-4613-3632-7_7. |
[7] |
J. Eckstein and W. Yao,
Approximate ADMM algorithms derived from Lagrangian splitting, Comput. Optim. Appl., 68 (2017), 363-405.
doi: 10.1007/s10589-017-9911-z. |
[8] |
J. Eckstein and W. Yao,
Relative-error approximate versions of Douglas-Rachford splitting and special cases of the ADMM, Math. Program., 170 (2018), 417-444.
doi: 10.1007/s10107-017-1160-5. |
[9] |
M. Fazel, T. K. Pong, D. Sun and P. Tseng,
Hankel matrix rank minimization with applications to system identification and realization, SIAM J. Matrix Anal. Appl., 34 (2013), 946-977.
doi: 10.1137/110853996. |
[10] |
R. Fletcher, Practical Methods of Optimization, Wiley-Interscience [John Wiley & Sons], New York, 2001. |
[11] |
C. A. Floudas and P. M. Pardalos, Encyclopedia of Optimization. Vol. I–VI, Kluwer Academic Publishers, Dordrecht, 2001.
doi: 10.1016/j.tcs.2009.07.038. |
[12] |
D. Gabay and B. Mercier,
A dual algorithm for the solution of nonlinear variational problems via finite element approximation, Computers & Mathematics with Applications, 2 (1976), 17-40.
doi: 10.1016/0898-1221(76)90003-1. |
[13] |
R. Glowinski and A. Marroco, Sur l'approximation, par éléments finis d'ordre un, et la résolution, par pénalisation-dualité, d'une classe de problèmes de Dirichlet non linéaires, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge Anal. Numér., 9 (1975), 41–76.
doi: 10.1051/m2an/197509R200411. |
[14] |
M. L. N. Gon{\c{c}}alves, M. M. Alves and J. G. Melo,
Pointwise and ergodic convergence rates of a variable metric proximal alternating direction method of multipliers, J. Optim. Theory Appl., 177 (2018), 448-478.
doi: 10.1007/s10957-018-1232-6. |
[15] |
Y. Gu and N. Yamashita, An alternating direction method of multipliers with the BFGS update for structured convex quadratic optimization, preprint, arXiv: 1903.02270. Google Scholar |
[16] |
B. He, L.-Z. Liao, D. Han and H. Yang,
A new inexact alternating directions method for monotone variational inequalities, Math. Program., 92 (2002), 103-118.
doi: 10.1007/s101070100280. |
[17] |
B. He, F. Ma and X. Yuan,
Optimally linearizing the alternating direction method of multipliers for convex programming, Comput. Optim. Appl., 75 (2020), 361-388.
doi: 10.1007/s10589-019-00152-3. |
[18] |
B. He and X. Yuan,
On the ${O}(1/n)$ convergence rate of the {D}ouglas-{R}achford alternating direction method, SIAM J. Numer. Anal., 50 (2012), 700-709.
doi: 10.1137/110836936. |
[19] |
K. Koh, S.-J. Kim and S. Boyd,
An interior-point method for large-scale $l_1$-regularized logistic regression, J. Mach. Learn. Res., 8 (2007), 1519-1555.
|
[20] |
M. Li, D. Sun and K.-C. Toh,
A majorized ADMM with indefinite proximal terms for linearly constrained convex composite optimization, SIAM J. Optim., 26 (2016), 922-950.
doi: 10.1137/140999025. |
[21] |
P. A. Lotito, L. A. Parente and M. V. Solodov,
A class of variable metric decomposition methods for monotone variational inclusions, J. Convex Anal., 16 (2009), 857-880.
|
[22] |
D. G. Luenberger and Y. Ye, Linear and nonlinear programming, in International Series in Operations Research & Management Science, 228, Springer, Cham, 1984.
doi: 10.1007/978-3-319-18842-3. |
[23] |
J. Nocedal and S. J. Wright, Numerical Optimization, Springer Series in Operations Research and Financial Engineering, 2$^nd$ edition, Springer, New York, 2006. |
[24] |
R. T. Rockafellar and R. J.-B. Wets, Numerical Optimization, Springer Science & Business Media, 2009. Google Scholar |
[25] |
M. H. Xu and T. Wu,
A class of linearized proximal alternating direction methods, J. Optim. Theory Appl., 151 (2011), 321-337.
doi: 10.1007/s10957-011-9876-5. |
[26] |
X.-M. Yuan,
The improvement with relative errors of He et al.'s inexact alternating direction method for monotone variational inequalities, Math. Comput. Modelling, 42 (2005), 1225-1236.
doi: 10.1016/j.mcm.2005.04.007. |
show all references
References:
[1] |
S. Banert, R. I. Bot and E. R. Csetnek, Fixing and extending some recent results on the ADMM algorithm, preprint, arXiv: 1612.05057. Google Scholar |
[2] |
S. Boyd, N. Parikh, E. Chu, B. Peleato and J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers, Foundations and TrendsⓇ in Machine Learning, 3 (2011), 1–122. Google Scholar |
[3] |
G. Chen and M. Teboulle,
A proximal-based decomposition method for convex minimization problems, Math. Programming, 64 (1994), 81-101.
doi: 10.1007/BF01582566. |
[4] |
W. Deng and W. Yin,
On the global and linear convergence of the generalized alternating direction method of multipliers, J. Sci. Comput., 66 (2016), 889-916.
doi: 10.1007/s10915-015-0048-x. |
[5] |
J. Eckstein and D. P. Bertsekas,
On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators, Math. Programming, 55 (1992), 293-318.
doi: 10.1007/BF01581204. |
[6] |
J. Eckstein and M. Fukushima, Some reformulations and applications of the alternating direction method of multipliers, in Large Scale Optimization (Gainesville, FL, 1993), Kluwer Acad. Publ., Dordrecht, 1994,115–134.
doi: 10.1007/978-1-4613-3632-7_7. |
[7] |
J. Eckstein and W. Yao,
Approximate ADMM algorithms derived from Lagrangian splitting, Comput. Optim. Appl., 68 (2017), 363-405.
doi: 10.1007/s10589-017-9911-z. |
[8] |
J. Eckstein and W. Yao,
Relative-error approximate versions of Douglas-Rachford splitting and special cases of the ADMM, Math. Program., 170 (2018), 417-444.
doi: 10.1007/s10107-017-1160-5. |
[9] |
M. Fazel, T. K. Pong, D. Sun and P. Tseng,
Hankel matrix rank minimization with applications to system identification and realization, SIAM J. Matrix Anal. Appl., 34 (2013), 946-977.
doi: 10.1137/110853996. |
[10] |
R. Fletcher, Practical Methods of Optimization, Wiley-Interscience [John Wiley & Sons], New York, 2001. |
[11] |
C. A. Floudas and P. M. Pardalos, Encyclopedia of Optimization. Vol. I–VI, Kluwer Academic Publishers, Dordrecht, 2001.
doi: 10.1016/j.tcs.2009.07.038. |
[12] |
D. Gabay and B. Mercier,
A dual algorithm for the solution of nonlinear variational problems via finite element approximation, Computers & Mathematics with Applications, 2 (1976), 17-40.
doi: 10.1016/0898-1221(76)90003-1. |
[13] |
R. Glowinski and A. Marroco, Sur l'approximation, par éléments finis d'ordre un, et la résolution, par pénalisation-dualité, d'une classe de problèmes de Dirichlet non linéaires, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge Anal. Numér., 9 (1975), 41–76.
doi: 10.1051/m2an/197509R200411. |
[14] |
M. L. N. Gon{\c{c}}alves, M. M. Alves and J. G. Melo,
Pointwise and ergodic convergence rates of a variable metric proximal alternating direction method of multipliers, J. Optim. Theory Appl., 177 (2018), 448-478.
doi: 10.1007/s10957-018-1232-6. |
[15] |
Y. Gu and N. Yamashita, An alternating direction method of multipliers with the BFGS update for structured convex quadratic optimization, preprint, arXiv: 1903.02270. Google Scholar |
[16] |
B. He, L.-Z. Liao, D. Han and H. Yang,
A new inexact alternating directions method for monotone variational inequalities, Math. Program., 92 (2002), 103-118.
doi: 10.1007/s101070100280. |
[17] |
B. He, F. Ma and X. Yuan,
Optimally linearizing the alternating direction method of multipliers for convex programming, Comput. Optim. Appl., 75 (2020), 361-388.
doi: 10.1007/s10589-019-00152-3. |
[18] |
B. He and X. Yuan,
On the ${O}(1/n)$ convergence rate of the {D}ouglas-{R}achford alternating direction method, SIAM J. Numer. Anal., 50 (2012), 700-709.
doi: 10.1137/110836936. |
[19] |
K. Koh, S.-J. Kim and S. Boyd,
An interior-point method for large-scale $l_1$-regularized logistic regression, J. Mach. Learn. Res., 8 (2007), 1519-1555.
|
[20] |
M. Li, D. Sun and K.-C. Toh,
A majorized ADMM with indefinite proximal terms for linearly constrained convex composite optimization, SIAM J. Optim., 26 (2016), 922-950.
doi: 10.1137/140999025. |
[21] |
P. A. Lotito, L. A. Parente and M. V. Solodov,
A class of variable metric decomposition methods for monotone variational inclusions, J. Convex Anal., 16 (2009), 857-880.
|
[22] |
D. G. Luenberger and Y. Ye, Linear and nonlinear programming, in International Series in Operations Research & Management Science, 228, Springer, Cham, 1984.
doi: 10.1007/978-3-319-18842-3. |
[23] |
J. Nocedal and S. J. Wright, Numerical Optimization, Springer Series in Operations Research and Financial Engineering, 2$^nd$ edition, Springer, New York, 2006. |
[24] |
R. T. Rockafellar and R. J.-B. Wets, Numerical Optimization, Springer Science & Business Media, 2009. Google Scholar |
[25] |
M. H. Xu and T. Wu,
A class of linearized proximal alternating direction methods, J. Optim. Theory Appl., 151 (2011), 321-337.
doi: 10.1007/s10957-011-9876-5. |
[26] |
X.-M. Yuan,
The improvement with relative errors of He et al.'s inexact alternating direction method for monotone variational inequalities, Math. Comput. Modelling, 42 (2005), 1225-1236.
doi: 10.1016/j.mcm.2005.04.007. |

Setting | ADM-Broyden |
ADM-Broyden |
ADM-Broyden |
|||||||||||
Iter. | Int.Iter. | Time | Iter. | Int.Iter. | Time | Iter. | Int.Iter. | Time | ||||||
500 | 200 | 0.1 | 2.4 | 72.0 | 216.0 | 0.16 | 2.3 | 74.0 | 222.0 | 0.16 | 2.4 | 72.0 | 216.0 | 0.16 |
500 | 500 | 0.1 | 3.3 | 81.0 | 243.0 | 0.46 | 3.5 | 83.0 | 249.0 | 0.49 | 2.2 | 82.0 | 246.0 | 0.44 |
1000 | 500 | 0.1 | 2.3 | 158.0 | 474.0 | 1.14 | 2.2 | 155.0 | 465.0 | 1.11 | 2.2 | 157.0 | 471.0 | 1.12 |
Setting | ADM-Broyden |
ADM-Broyden |
ADM-Broyden |
|||||||||||
Iter. | Int.Iter. | Time | Iter. | Int.Iter. | Time | Iter. | Int.Iter. | Time | ||||||
500 | 200 | 0.1 | 2.4 | 72.0 | 216.0 | 0.16 | 2.3 | 74.0 | 222.0 | 0.16 | 2.4 | 72.0 | 216.0 | 0.16 |
500 | 500 | 0.1 | 3.3 | 81.0 | 243.0 | 0.46 | 3.5 | 83.0 | 249.0 | 0.49 | 2.2 | 82.0 | 246.0 | 0.44 |
1000 | 500 | 0.1 | 2.3 | 158.0 | 474.0 | 1.14 | 2.2 | 155.0 | 465.0 | 1.11 | 2.2 | 157.0 | 471.0 | 1.12 |
Setting | ADMM-1 | ADMM-2 | ADM-PRO | ADM-BFGS | ||||||||||||||
Iter. | Int.Iter. | Time | Iter. | Int.Iter. | Time | Iter. | Int.Iter. | Time | Iter. | Int.Iter. | Time | |||||||
500 | 200 | 0.1 | 4.0 | 15.0 | 60.0 | 0.35 | 0.6 | 77.0 | 306.0 | 0.14 | 0.3 | 131.0 | 524.0 | 0.23 | 2.3 | 74.0 | 222.0 | 0.16 |
500 | 500 | 0.1 | 4.0 | 20.0 | 100.0 | 2.50 | 0.6 | 105.0 | 418.0 | 0.26 | 0.3 | 390.0 | 1559.0 | 0.75 | 3.5 | 83.0 | 249.0 | 0.49 |
1000 | 500 | 0.1 | 9.0 | 19.0 | 85.0 | 4.59 | 0.8 | 128.0 | 510.0 | 0.51 | 0.3 | 233.0 | 931.0 | 0.88 | 2.2 | 155.0 | 465.0 | 1.11 |
Setting | ADMM-1 | ADMM-2 | ADM-PRO | ADM-BFGS | ||||||||||||||
Iter. | Int.Iter. | Time | Iter. | Int.Iter. | Time | Iter. | Int.Iter. | Time | Iter. | Int.Iter. | Time | |||||||
500 | 200 | 0.1 | 4.0 | 15.0 | 60.0 | 0.35 | 0.6 | 77.0 | 306.0 | 0.14 | 0.3 | 131.0 | 524.0 | 0.23 | 2.3 | 74.0 | 222.0 | 0.16 |
500 | 500 | 0.1 | 4.0 | 20.0 | 100.0 | 2.50 | 0.6 | 105.0 | 418.0 | 0.26 | 0.3 | 390.0 | 1559.0 | 0.75 | 3.5 | 83.0 | 249.0 | 0.49 |
1000 | 500 | 0.1 | 9.0 | 19.0 | 85.0 | 4.59 | 0.8 | 128.0 | 510.0 | 0.51 | 0.3 | 233.0 | 931.0 | 0.88 | 2.2 | 155.0 | 465.0 | 1.11 |
Algorithm | ||||||||||||||||||
Iter. | Int.Iter. | Time | T-A | T-M | Iter. | Int.Iter. | Time | T-A | T-M | Iter. | Int.Iter. | Time | T-A | T-M | ||||
ADMM-1 | 9.0 | 17.0 | 85.0 | 4.59 | 10.0 | 19.0 | 95.0 | 13.46 | 9.0 | 23.0 | 115.0 | 53.47 | ||||||
[3pt] ADMM-2 | 0.8 | 128.0 | 510.0 | 0.51 | 0.46 | 0.05 | 0.8 | 159.0 | 633.0 | 1.18 | 0.99 | 0.19 | 0.6 | 223.0 | 891.0 | 3.13 | 2.93 | 0.20 |
[3pt] ADM-PRO | 0.3 | 233.0 | 931.0 | 0.86 | 0.83 | 0.03 | 0.4 | 550.0 | 2199.0 | 2.64 | 2.59 | 0.05 | 0.2 | 941.0 | 3764.0 | 10.68 | 10.61 | 0.07 |
[3pt] ADM-LBFGS | 0.7 | 143.0 | 570.0 | 0.55 | 0.52 | 0.03 | 0.8 | 184.0 | 733.0 | 0.94 | 0.89 | 0.05 | 0.8 | 305.0 | 1217.0 | 3.75 | 3.68 | 0.07 |
[3pt] ADM-ILBFGS | 0.7 | 139.0 | 554.0 | 0.55 | 0.52 | 0.03 | 1.0 | 185.0 | 736.0 | 0.95 | 0.90 | 0.05 | 0.9 | 294.0 | 1173.0 | 3.58 | 3.51 | 0.07 |
Iter. | Int.Iter. | Time | T-A | T-M | Iter. | Int.Iter. | Time | T-A | T-M | Iter. | Int.Iter. | Time | T-A | T-M | ||||
ADMM-1 | 40 | 16.0 | 80.0 | 153.79 | 161 | 15.0 | 74.0 | 612.32 | 250 | 15.0 | 74.0 | 984.23 | ||||||
[3pt] ADMM-2 | 1.7 | 266.0 | 1057.0 | 11.67 | 11.41 | 0.26 | 3.9 | 526.0 | 1578.0 | 81.18 | 79.56 | 1.62 | 4.6 | 665.0 | 1995.0 | 142.19 | 138.47 | 3.72 |
[3pt] ADM-PRO | 1.1 | 344.0 | 1373.0 | 14.40 | 14.19 | 0.12 | 2.1 | 719.0 | 2859.0 | 111.96 | 111.41 | 0.55 | 2.6 | 920.0 | 3625.0 | 196.81 | 195.91 | 0.90 |
[3pt] ADM-LBFGS | 1.7 | 268.0 | 1064.0 | 11.22 | 11.10 | 0.12 | 3.7 | 531.0 | 1593.0 | 80.30 | 79.75 | 0.55 | 4.6 | 669.0 | 2007.0 | 142.71 | 141.81 | 0.90 |
[3pt] ADM-ILBFGS | 1.7 | 267.0 | 1061.0 | 11.02 | 10.90 | 0.12 | 3.8 | 528.0 | 1584.0 | 79.80 | 79.25 | 0.55 | 4.6 | 667.0 | 2001.0 | 141.63 | 140.73 | 0.90 |
Iter. | Int.Iter. | Time | T-A | T-M | Iter. | Int.Iter. | Time | T-A | T-M | Iter. | Int.Iter. | Time | T-A | T-M | ||||
ADMM-2 | 2.2 | 475.0 | 1892.0 | 193.10 | 181.67 | 11.43 | 4.3 | 1005.0 | 3015.0 | 1407.36 | 1316.69 | 90.67 | 5.3 | 1258.0 | 3774.0 | 2769.45 | 2554.13 | 215.32 |
[5pt] ADM-PRO | 0.6 | 736.0 | 2944.0 | 262.77 | 259.87 | 2.90 | 0.9 | 1405.0 | 5618.0 | 1843.21 | 1834.80 | 8.41 | 1.0 | 1780.0 | 7118.0 | 3664.39 | 3653.84 | 10.55 |
[5pt] ADM-LBFGS | 2.2 | 486.0 | 1935.0 | 178.95 | 176.05 | 2.90 | 4.0 | 1038.0 | 3114.0 | 1370.20 | 1361.79 | 8.41 | 4.5 | 1319.0 | 3957.0 | 2712.25 | 2701.70 | 10.55 |
[5pt] ADM-ILBFGS | 2.3 | 477.0 | 1898.0 | 175.98 | 173.08 | 2.90 | 4.2 | 1024.0 | 3072.0 | 1359.35 | 1350.94 | 8.41 | 5.0 | 1278.0 | 3834.0 | 2629.89 | 2619.34 | 10.55 |
Algorithm | ||||||||||||||||||
Iter. | Int.Iter. | Time | T-A | T-M | Iter. | Int.Iter. | Time | T-A | T-M | Iter. | Int.Iter. | Time | T-A | T-M | ||||
ADMM-1 | 9.0 | 17.0 | 85.0 | 4.59 | 10.0 | 19.0 | 95.0 | 13.46 | 9.0 | 23.0 | 115.0 | 53.47 | ||||||
[3pt] ADMM-2 | 0.8 | 128.0 | 510.0 | 0.51 | 0.46 | 0.05 | 0.8 | 159.0 | 633.0 | 1.18 | 0.99 | 0.19 | 0.6 | 223.0 | 891.0 | 3.13 | 2.93 | 0.20 |
[3pt] ADM-PRO | 0.3 | 233.0 | 931.0 | 0.86 | 0.83 | 0.03 | 0.4 | 550.0 | 2199.0 | 2.64 | 2.59 | 0.05 | 0.2 | 941.0 | 3764.0 | 10.68 | 10.61 | 0.07 |
[3pt] ADM-LBFGS | 0.7 | 143.0 | 570.0 | 0.55 | 0.52 | 0.03 | 0.8 | 184.0 | 733.0 | 0.94 | 0.89 | 0.05 | 0.8 | 305.0 | 1217.0 | 3.75 | 3.68 | 0.07 |
[3pt] ADM-ILBFGS | 0.7 | 139.0 | 554.0 | 0.55 | 0.52 | 0.03 | 1.0 | 185.0 | 736.0 | 0.95 | 0.90 | 0.05 | 0.9 | 294.0 | 1173.0 | 3.58 | 3.51 | 0.07 |
Iter. | Int.Iter. | Time | T-A | T-M | Iter. | Int.Iter. | Time | T-A | T-M | Iter. | Int.Iter. | Time | T-A | T-M | ||||
ADMM-1 | 40 | 16.0 | 80.0 | 153.79 | 161 | 15.0 | 74.0 | 612.32 | 250 | 15.0 | 74.0 | 984.23 | ||||||
[3pt] ADMM-2 | 1.7 | 266.0 | 1057.0 | 11.67 | 11.41 | 0.26 | 3.9 | 526.0 | 1578.0 | 81.18 | 79.56 | 1.62 | 4.6 | 665.0 | 1995.0 | 142.19 | 138.47 | 3.72 |
[3pt] ADM-PRO | 1.1 | 344.0 | 1373.0 | 14.40 | 14.19 | 0.12 | 2.1 | 719.0 | 2859.0 | 111.96 | 111.41 | 0.55 | 2.6 | 920.0 | 3625.0 | 196.81 | 195.91 | 0.90 |
[3pt] ADM-LBFGS | 1.7 | 268.0 | 1064.0 | 11.22 | 11.10 | 0.12 | 3.7 | 531.0 | 1593.0 | 80.30 | 79.75 | 0.55 | 4.6 | 669.0 | 2007.0 | 142.71 | 141.81 | 0.90 |
[3pt] ADM-ILBFGS | 1.7 | 267.0 | 1061.0 | 11.02 | 10.90 | 0.12 | 3.8 | 528.0 | 1584.0 | 79.80 | 79.25 | 0.55 | 4.6 | 667.0 | 2001.0 | 141.63 | 140.73 | 0.90 |
Iter. | Int.Iter. | Time | T-A | T-M | Iter. | Int.Iter. | Time | T-A | T-M | Iter. | Int.Iter. | Time | T-A | T-M | ||||
ADMM-2 | 2.2 | 475.0 | 1892.0 | 193.10 | 181.67 | 11.43 | 4.3 | 1005.0 | 3015.0 | 1407.36 | 1316.69 | 90.67 | 5.3 | 1258.0 | 3774.0 | 2769.45 | 2554.13 | 215.32 |
[5pt] ADM-PRO | 0.6 | 736.0 | 2944.0 | 262.77 | 259.87 | 2.90 | 0.9 | 1405.0 | 5618.0 | 1843.21 | 1834.80 | 8.41 | 1.0 | 1780.0 | 7118.0 | 3664.39 | 3653.84 | 10.55 |
[5pt] ADM-LBFGS | 2.2 | 486.0 | 1935.0 | 178.95 | 176.05 | 2.90 | 4.0 | 1038.0 | 3114.0 | 1370.20 | 1361.79 | 8.41 | 4.5 | 1319.0 | 3957.0 | 2712.25 | 2701.70 | 10.55 |
[5pt] ADM-ILBFGS | 2.3 | 477.0 | 1898.0 | 175.98 | 173.08 | 2.90 | 4.2 | 1024.0 | 3072.0 | 1359.35 | 1350.94 | 8.41 | 5.0 | 1278.0 | 3834.0 | 2629.89 | 2619.34 | 10.55 |
[1] |
Jie-Wen He, Chi-Chon Lei, Chen-Yang Shi, Seak-Weng Vong. An inexact alternating direction method of multipliers for a kind of nonlinear complementarity problems. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 353-362. doi: 10.3934/naco.2020030 |
[2] |
Tengteng Yu, Xin-Wei Liu, Yu-Hong Dai, Jie Sun. Variable metric proximal stochastic variance reduced gradient methods for nonconvex nonsmooth optimization. Journal of Industrial & Management Optimization, 2021 doi: 10.3934/jimo.2021084 |
[3] |
Mohsen Abdolhosseinzadeh, Mir Mohammad Alipour. Design of experiment for tuning parameters of an ant colony optimization method for the constrained shortest Hamiltonian path problem in the grid networks. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 321-332. doi: 10.3934/naco.2020028 |
[4] |
Mostafa Ghelichi, A. M. Goltabar, H. R. Tavakoli, A. Karamodin. Neuro-fuzzy active control optimized by Tug of war optimization method for seismically excited benchmark highway bridge. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 333-351. doi: 10.3934/naco.2020029 |
[5] |
Kazeem Olalekan Aremu, Chinedu Izuchukwu, Grace Nnenanya Ogwo, Oluwatosin Temitope Mewomo. Multi-step iterative algorithm for minimization and fixed point problems in p-uniformly convex metric spaces. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2161-2180. doi: 10.3934/jimo.2020063 |
[6] |
Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271 |
[7] |
Bingru Zhang, Chuanye Gu, Jueyou Li. Distributed convex optimization with coupling constraints over time-varying directed graphs†. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2119-2138. doi: 10.3934/jimo.2020061 |
[8] |
Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327 |
[9] |
Mikhail Dokuchaev, Guanglu Zhou, Song Wang. A modification of Galerkin's method for option pricing. Journal of Industrial & Management Optimization, 2021 doi: 10.3934/jimo.2021077 |
[10] |
Jiahui Chen, Rundong Zhao, Yiying Tong, Guo-Wei Wei. Evolutionary de Rham-Hodge method. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3785-3821. doi: 10.3934/dcdsb.2020257 |
[11] |
Bing Gao, Rui Gao. On fair entropy of the tent family. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3797-3816. doi: 10.3934/dcds.2021017 |
[12] |
Tao Wu, Yu Lei, Jiao Shi, Maoguo Gong. An evolutionary multiobjective method for low-rank and sparse matrix decomposition. Big Data & Information Analytics, 2017, 2 (1) : 23-37. doi: 10.3934/bdia.2017006 |
[13] |
Deren Han, Zehui Jia, Yongzhong Song, David Z. W. Wang. An efficient projection method for nonlinear inverse problems with sparsity constraints. Inverse Problems & Imaging, 2016, 10 (3) : 689-709. doi: 10.3934/ipi.2016017 |
[14] |
Boris Kramer, John R. Singler. A POD projection method for large-scale algebraic Riccati equations. Numerical Algebra, Control & Optimization, 2016, 6 (4) : 413-435. doi: 10.3934/naco.2016018 |
[15] |
Petra Csomós, Hermann Mena. Fourier-splitting method for solving hyperbolic LQR problems. Numerical Algebra, Control & Optimization, 2018, 8 (1) : 17-46. doi: 10.3934/naco.2018002 |
[16] |
Christina Surulescu, Nicolae Surulescu. Modeling and simulation of some cell dispersion problems by a nonparametric method. Mathematical Biosciences & Engineering, 2011, 8 (2) : 263-277. doi: 10.3934/mbe.2011.8.263 |
[17] |
Jiangxing Wang. Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2429-2440. doi: 10.3934/dcdsb.2020185 |
[18] |
Christoforidou Amalia, Christian-Oliver Ewald. A lattice method for option evaluation with regime-switching asset correlation structure. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1729-1752. doi: 10.3934/jimo.2020042 |
[19] |
Hong-Yi Miao, Li Wang. Preconditioned inexact Newton-like method for large nonsymmetric eigenvalue problems. Numerical Algebra, Control & Optimization, 2021 doi: 10.3934/naco.2021012 |
[20] |
Xiaofei Liu, Yong Wang. Weakening convergence conditions of a potential reduction method for tensor complementarity problems. Journal of Industrial & Management Optimization, 2021 doi: 10.3934/jimo.2021080 |
2019 Impact Factor: 1.366
Tools
Metrics
Other articles
by authors
[Back to Top]