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Alternating direction method of multipliers with variable metric indefinite proximal terms for convex optimization
1. | Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan |
This paper studies a proximal alternating direction method of multipliers (ADMM) with variable metric indefinite proximal terms for linearly constrained convex optimization problems. The proximal ADMM plays an important role in many application areas, since the subproblems of the method are easy to solve. Recently, it is reported that the proximal ADMM with a certain fixed indefinite proximal term is faster than that with a positive semidefinite term, and still has the global convergence property. On the other hand, Gu and Yamashita studied a variable metric semidefinite proximal ADMM whose proximal term is generated by the BFGS update. They reported that a slightly indefinite matrix also makes the algorithm work well in their numerical experiments. Motivated by this fact, we consider a variable metric indefinite proximal ADMM, and give sufficient conditions on the proximal terms for the global convergence. Moreover, based on the BFGS update, we propose a new indefinite proximal term which can satisfy the conditions for the global convergence. Experiments on several datasets demonstrated that our proposed variable metric indefinite proximal ADMM outperforms most of the comparison proximal ADMMs.
References:
[1] |
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.
doi: 10.1561/2200000016. |
[2] |
A. Chambolle and T. Pock,
A first-order primal-dual algorithm for convex problems with applications to imaging, Journal of Mathematical Imaging and Vision, 40 (2011), 120-145.
doi: 10.1007/s10851-010-0251-1. |
[3] |
W. Deng and W. Yin,
On the global and linear convergence of the generalized alternating direction method of multipliers, Journal of Scientific Computing, 66 (2016), 889-916.
doi: 10.1007/s10915-015-0048-x. |
[4] |
J. Eckstein,
Some saddle-function splitting methods for convex programming, Optimization Methods and Software, 4 (1994), 75-83.
doi: 10.1080/10556789408805578. |
[5] |
J. Eckstein and D. P. Bertsekas,
On the douglas-rachford splitting method and the proximal point algorithm for maximal monotone operators, Mathematical Programming, 55 (1992), 293-318.
doi: 10.1007/BF01581204. |
[6] |
J. Eckstein and W. Yao,
Relative-error approximate versions of Douglas–Rachford splitting and special cases of the ADMM, Mathematical Programming, 170 (2018), 417-444.
doi: 10.1007/s10107-017-1160-5. |
[7] |
M. Fazel, T. K. Pong, D. Sun and P. Tseng,
Hankel matrix rank minimization with applications to system identification and realization, SIAM Journal on Matrix Analysis and Applications, 34 (2013), 946-977.
doi: 10.1137/110853996. |
[8] |
M. Fortin and R. Glowinski, Chapter ⅲ on decomposition-coordination methods using an augmented lagrangian, in Studies in Mathematics and Its Applications, vol. 15, Elsevier, (1983), 97–146.
doi: 10.1016/S0168-2024(08)70028-6. |
[9] |
D. Gabay and B. Mercier,
A dual algorithm for the solution of nonlinear variational problems via finite element approximation, Computers and Mathematics with Applications, 2 (1976), 17-40.
doi: 10.1016/0898-1221(76)90003-1. |
[10] |
B. Gao and F. Ma,
Symmetric alternating direction method with indefinite proximal regularization for linearly constrained convex optimization, Journal of Optimization Theory and Applications, 176 (2018), 178-204.
doi: 10.1007/s10957-017-1207-z. |
[11] |
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, ESAIM: Mathematical Modelling and Numerical Analysis-Modélisation Mathématique et Analyse Numérique, 9 (1975), 41–76.
doi: 10.1051/m2an/197509R200411. |
[12] |
M. L. N. Gonçalves, M. M. Alves and J. G. Melo,
Pointwise and ergodic convergence rates of a variable metric proximal alternating direction method of multipliers, Journal of Optimization Theory and Applications, 177 (2018), 448-478.
doi: 10.1007/s10957-018-1232-6. |
[13] |
Y. Gu and N. Yamashita, An alternating direction method of multipliers with the BFGS update for structured convex quadratic optimization, preprint, arXiv: 1903.02270. |
[14] |
Y. Gu and N. Yamashita, A proximal ADMM with the Broyden family for convex optimization problems, Journal of Industrial and Management Optimization, 13 (2020).
doi: 10.3934/jimo.2020091. |
[15] |
D. Harrison Jr and D. L. Rubinfeld,
Hedonic housing prices and the demand for clean air, Journal of Environmental Economics and Management, 5 (1978), 81-102.
doi: 10.1016/0095-0696(78)90006-2. |
[16] |
B. He, L.-Z. Liao, D. Han and H. Yang,
A new inexact alternating directions method for monotone variational inequalities, Mathematical Programming, 92 (2002), 103-118.
doi: 10.1007/s101070100280. |
[17] |
B. He, H. Liu, Z. Wang and X. Yuan,
A strictly contractive Peaceman–Rachford splitting method for convex programming, SIAM Journal on Optimization, 24 (2014), 1011-1040.
doi: 10.1137/13090849X. |
[18] |
B. He and X. Yuan,
On the $O(1/n)$ convergence rate of the Douglas-Rachford alternating direction method, SIAM Journal on Numerical Analysis, 50 (2012), 700-709.
doi: 10.1137/110836936. |
[19] |
B. He and X. Yuan,
Block-wise alternating direction method of multipliers for multiple-block convex programming and beyond, SMAI Journal of Computational Mathematics, 1 (2015), 145-174.
doi: 10.5802/smai-jcm.6. |
[20] |
B. He, F. Ma and X. Yuan, Optimally linearizing the alternating direction method of multipliers for convex programming, Computational Optimization and Applications, (2019), 1–28.
doi: 10.1007/s10589-019-00152-3. |
[21] |
K. Koh, S.-J. Kim and S. Boyd,
An interior-point method for large-scale l1-regularized logistic regression, Journal of Machine Learning Research, 8 (2007), 1519-1555.
doi: 10.1109/JSTSP.2007.910971. |
[22] |
M. Li, D. Sun and K.-C. Toh,
A majorized admm with indefinite proximal terms for linearly constrained convex composite optimization, SIAM Journal on Optimization, 26 (2016), 922-950.
doi: 10.1137/140999025. |
[23] |
P.-L. Lions and B. Mercier,
Splitting algorithms for the sum of two nonlinear operators, SIAM Journal on Numerical Analysis, 16 (1979), 964-979.
doi: 10.1137/0716071. |
[24] |
P. A. Lotito, L. A. Parente and M. Solodov,
A class of variable metric decomposition methods for monotone variational inclusions, Journal of Convex Analysis, 16 (2009), 857-880.
|
[25] |
Y. Nesterov,
Gradient methods for minimizing composite functions, Mathematical Programming, 140 (2013), 125-161.
doi: 10.1007/s10107-012-0629-5. |
[26] |
R. K. Pace and R. Barry,
Sparse spatial autoregressions, Statistics and Probability Letters, 33 (1997), 291-297.
doi: 10.1016/S0167-7152(96)00140-X. |
[27] |
L. I. Rudin, S. Osher and E. Fatemi,
Nonlinear total variation based noise removal algorithms, Physica D: Nonlinear Phenomena, 60 (1992), 259-268.
doi: 10.1016/0167-2789(92)90242-F. |
[28] |
H. Trevor, T. Robert and F. JH, The Elements of Statistical Learning: Data Mining, Inference, and Prediction, Springer Science and Business Media, 2009. |
[29] |
P. Tseng and S. Yun,
A coordinate gradient descent method for nonsmooth separable minimization, Mathematical Programming, 117 (2009), 387-423.
doi: 10.1007/s10107-007-0170-0. |
[30] |
M. Xu,
Proximal alternating directions method for structured variational inequalities, Journal of Optimization Theory and Applications, 134 (2007), 107-117.
doi: 10.1007/s10957-007-9192-2. |
[31] |
M. Xu and T. Wu,
A class of linearized proximal alternating direction methods, Journal of Optimization Theory and Applications, 151 (2011), 321-337.
doi: 10.1007/s10957-011-9876-5. |
[32] |
W. Yin, S. Osher, D. Goldfarb and J. Darbon,
Bregman iterative algorithms for $l_1$-minimization with applications to compressed sensing, SIAM Journal on Imaging Sciences, 1 (2008), 143-168.
doi: 10.1137/070703983. |
show all references
References:
[1] |
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.
doi: 10.1561/2200000016. |
[2] |
A. Chambolle and T. Pock,
A first-order primal-dual algorithm for convex problems with applications to imaging, Journal of Mathematical Imaging and Vision, 40 (2011), 120-145.
doi: 10.1007/s10851-010-0251-1. |
[3] |
W. Deng and W. Yin,
On the global and linear convergence of the generalized alternating direction method of multipliers, Journal of Scientific Computing, 66 (2016), 889-916.
doi: 10.1007/s10915-015-0048-x. |
[4] |
J. Eckstein,
Some saddle-function splitting methods for convex programming, Optimization Methods and Software, 4 (1994), 75-83.
doi: 10.1080/10556789408805578. |
[5] |
J. Eckstein and D. P. Bertsekas,
On the douglas-rachford splitting method and the proximal point algorithm for maximal monotone operators, Mathematical Programming, 55 (1992), 293-318.
doi: 10.1007/BF01581204. |
[6] |
J. Eckstein and W. Yao,
Relative-error approximate versions of Douglas–Rachford splitting and special cases of the ADMM, Mathematical Programming, 170 (2018), 417-444.
doi: 10.1007/s10107-017-1160-5. |
[7] |
M. Fazel, T. K. Pong, D. Sun and P. Tseng,
Hankel matrix rank minimization with applications to system identification and realization, SIAM Journal on Matrix Analysis and Applications, 34 (2013), 946-977.
doi: 10.1137/110853996. |
[8] |
M. Fortin and R. Glowinski, Chapter ⅲ on decomposition-coordination methods using an augmented lagrangian, in Studies in Mathematics and Its Applications, vol. 15, Elsevier, (1983), 97–146.
doi: 10.1016/S0168-2024(08)70028-6. |
[9] |
D. Gabay and B. Mercier,
A dual algorithm for the solution of nonlinear variational problems via finite element approximation, Computers and Mathematics with Applications, 2 (1976), 17-40.
doi: 10.1016/0898-1221(76)90003-1. |
[10] |
B. Gao and F. Ma,
Symmetric alternating direction method with indefinite proximal regularization for linearly constrained convex optimization, Journal of Optimization Theory and Applications, 176 (2018), 178-204.
doi: 10.1007/s10957-017-1207-z. |
[11] |
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, ESAIM: Mathematical Modelling and Numerical Analysis-Modélisation Mathématique et Analyse Numérique, 9 (1975), 41–76.
doi: 10.1051/m2an/197509R200411. |
[12] |
M. L. N. Gonçalves, M. M. Alves and J. G. Melo,
Pointwise and ergodic convergence rates of a variable metric proximal alternating direction method of multipliers, Journal of Optimization Theory and Applications, 177 (2018), 448-478.
doi: 10.1007/s10957-018-1232-6. |
[13] |
Y. Gu and N. Yamashita, An alternating direction method of multipliers with the BFGS update for structured convex quadratic optimization, preprint, arXiv: 1903.02270. |
[14] |
Y. Gu and N. Yamashita, A proximal ADMM with the Broyden family for convex optimization problems, Journal of Industrial and Management Optimization, 13 (2020).
doi: 10.3934/jimo.2020091. |
[15] |
D. Harrison Jr and D. L. Rubinfeld,
Hedonic housing prices and the demand for clean air, Journal of Environmental Economics and Management, 5 (1978), 81-102.
doi: 10.1016/0095-0696(78)90006-2. |
[16] |
B. He, L.-Z. Liao, D. Han and H. Yang,
A new inexact alternating directions method for monotone variational inequalities, Mathematical Programming, 92 (2002), 103-118.
doi: 10.1007/s101070100280. |
[17] |
B. He, H. Liu, Z. Wang and X. Yuan,
A strictly contractive Peaceman–Rachford splitting method for convex programming, SIAM Journal on Optimization, 24 (2014), 1011-1040.
doi: 10.1137/13090849X. |
[18] |
B. He and X. Yuan,
On the $O(1/n)$ convergence rate of the Douglas-Rachford alternating direction method, SIAM Journal on Numerical Analysis, 50 (2012), 700-709.
doi: 10.1137/110836936. |
[19] |
B. He and X. Yuan,
Block-wise alternating direction method of multipliers for multiple-block convex programming and beyond, SMAI Journal of Computational Mathematics, 1 (2015), 145-174.
doi: 10.5802/smai-jcm.6. |
[20] |
B. He, F. Ma and X. Yuan, Optimally linearizing the alternating direction method of multipliers for convex programming, Computational Optimization and Applications, (2019), 1–28.
doi: 10.1007/s10589-019-00152-3. |
[21] |
K. Koh, S.-J. Kim and S. Boyd,
An interior-point method for large-scale l1-regularized logistic regression, Journal of Machine Learning Research, 8 (2007), 1519-1555.
doi: 10.1109/JSTSP.2007.910971. |
[22] |
M. Li, D. Sun and K.-C. Toh,
A majorized admm with indefinite proximal terms for linearly constrained convex composite optimization, SIAM Journal on Optimization, 26 (2016), 922-950.
doi: 10.1137/140999025. |
[23] |
P.-L. Lions and B. Mercier,
Splitting algorithms for the sum of two nonlinear operators, SIAM Journal on Numerical Analysis, 16 (1979), 964-979.
doi: 10.1137/0716071. |
[24] |
P. A. Lotito, L. A. Parente and M. Solodov,
A class of variable metric decomposition methods for monotone variational inclusions, Journal of Convex Analysis, 16 (2009), 857-880.
|
[25] |
Y. Nesterov,
Gradient methods for minimizing composite functions, Mathematical Programming, 140 (2013), 125-161.
doi: 10.1007/s10107-012-0629-5. |
[26] |
R. K. Pace and R. Barry,
Sparse spatial autoregressions, Statistics and Probability Letters, 33 (1997), 291-297.
doi: 10.1016/S0167-7152(96)00140-X. |
[27] |
L. I. Rudin, S. Osher and E. Fatemi,
Nonlinear total variation based noise removal algorithms, Physica D: Nonlinear Phenomena, 60 (1992), 259-268.
doi: 10.1016/0167-2789(92)90242-F. |
[28] |
H. Trevor, T. Robert and F. JH, The Elements of Statistical Learning: Data Mining, Inference, and Prediction, Springer Science and Business Media, 2009. |
[29] |
P. Tseng and S. Yun,
A coordinate gradient descent method for nonsmooth separable minimization, Mathematical Programming, 117 (2009), 387-423.
doi: 10.1007/s10107-007-0170-0. |
[30] |
M. Xu,
Proximal alternating directions method for structured variational inequalities, Journal of Optimization Theory and Applications, 134 (2007), 107-117.
doi: 10.1007/s10957-007-9192-2. |
[31] |
M. Xu and T. Wu,
A class of linearized proximal alternating direction methods, Journal of Optimization Theory and Applications, 151 (2011), 321-337.
doi: 10.1007/s10957-011-9876-5. |
[32] |
W. Yin, S. Osher, D. Goldfarb and J. Darbon,
Bregman iterative algorithms for $l_1$-minimization with applications to compressed sensing, SIAM Journal on Imaging Sciences, 1 (2008), 143-168.
doi: 10.1137/070703983. |



method | |||
Classical ADMM [9,11] | - | ||
Semi-proximal ADMM [7] | fixed positive semidefinite matrices | ||
Variable semi-proximal ADMM [13,14] | variable positive semidefinite matrices | 1 | |
Indefinite proximal ADMM [20] | fixed indefinite matrices | 1 | |
Proposed method | variable indefinite matrices | 1 |
method | |||
Classical ADMM [9,11] | - | ||
Semi-proximal ADMM [7] | fixed positive semidefinite matrices | ||
Variable semi-proximal ADMM [13,14] | variable positive semidefinite matrices | 1 | |
Indefinite proximal ADMM [20] | fixed indefinite matrices | 1 | |
Proposed method | variable indefinite matrices | 1 |
Construction of |
1 Let 2 Let 3 If 4 Otherwise 5 Construct |
Construction of |
1 Let 2 Let 3 If 4 Otherwise 5 Construct |
Datasets | Tolerance | ADMM | SPADM | IPADM | IADMB | IADML | ||||||
Iter. | T-A | Iter. | T-A | Iter. | T-A | Iter. | T-A | Iter. | T-A | |||
Boston-house |
100 | 22.0 | 0.0010 | 70.0 | 0.0015 | 48.0 | 0.0013 | 26.0 | 0.0015 | 25.0 | 0.0018 | |
300 | 9.0 | 0.0010 | 37.0 | 0.0013 | 32.0 | 0.0012 | 20.0 | 0.0015 | 21.0 | 0.0016 | ||
500 | 11.0 | 0.0010 | 29.0 | 0.0012 | 26.0 | 0.0011 | 20.0 | 0.0012 | 19.0 | 0.0014 | ||
300 | 16.0 | 0.0010 | 69.0 | 0.0014 | 60.0 | 0.0013 | 29.0 | 0.0014 | {34.0} | 0.0017 | ||
500 | 18.0 | 0.0011 | 75.0 | 0.0015 | 65.0 | 0.0014 | 28.0 | 0.0015 | 30.0 | 0.0016 | ||
700 | 25.0 | 0.0010 | 86.0 | 0.0017 | 75.0 | 0.0015 | 36.0 | 0.0017 | 36.0 | 0.0017 | ||
California-housing |
1000 | 36.0 | 0.0014 | 89.0 | 0.0069 | 77.0 | 0.0064 | 38.0 | 0.0022 | 38.0 | 0.0054 | |
1500 | 24.0 | 0.0012 | 73.0 | 0.0057 | 64.0 | 0.0056 | 26.0 | 0.0022 | 27.0 | 0.0050 | ||
1700 | 21.0 | 0.0013 | 67.0 | 0.0056 | 59.0 | 0.0049 | 24.0 | 0.0018 | 24.0 | 0.0041 | ||
5000 | 23.0 | 0.0013 | 55.0 | 0.0046 | 49.0 | 0.0044 | 28.0 | 0.0020 | 28.0 | 0.0048 | ||
5300 | 22.0 | 0.0014 | 53.0 | 0.0045 | 47.0 | 0.0043 | 27.0 | 0.0023 | 27.0 | 0.0042 | ||
5500 | 22.0 | 0.0012 | 52.0 | 0.0041 | 47.0 | 0.0042 | 26.0 | 0.0018 | 26.0 | 0.0036 |
Datasets | Tolerance | ADMM | SPADM | IPADM | IADMB | IADML | ||||||
Iter. | T-A | Iter. | T-A | Iter. | T-A | Iter. | T-A | Iter. | T-A | |||
Boston-house |
100 | 22.0 | 0.0010 | 70.0 | 0.0015 | 48.0 | 0.0013 | 26.0 | 0.0015 | 25.0 | 0.0018 | |
300 | 9.0 | 0.0010 | 37.0 | 0.0013 | 32.0 | 0.0012 | 20.0 | 0.0015 | 21.0 | 0.0016 | ||
500 | 11.0 | 0.0010 | 29.0 | 0.0012 | 26.0 | 0.0011 | 20.0 | 0.0012 | 19.0 | 0.0014 | ||
300 | 16.0 | 0.0010 | 69.0 | 0.0014 | 60.0 | 0.0013 | 29.0 | 0.0014 | {34.0} | 0.0017 | ||
500 | 18.0 | 0.0011 | 75.0 | 0.0015 | 65.0 | 0.0014 | 28.0 | 0.0015 | 30.0 | 0.0016 | ||
700 | 25.0 | 0.0010 | 86.0 | 0.0017 | 75.0 | 0.0015 | 36.0 | 0.0017 | 36.0 | 0.0017 | ||
California-housing |
1000 | 36.0 | 0.0014 | 89.0 | 0.0069 | 77.0 | 0.0064 | 38.0 | 0.0022 | 38.0 | 0.0054 | |
1500 | 24.0 | 0.0012 | 73.0 | 0.0057 | 64.0 | 0.0056 | 26.0 | 0.0022 | 27.0 | 0.0050 | ||
1700 | 21.0 | 0.0013 | 67.0 | 0.0056 | 59.0 | 0.0049 | 24.0 | 0.0018 | 24.0 | 0.0041 | ||
5000 | 23.0 | 0.0013 | 55.0 | 0.0046 | 49.0 | 0.0044 | 28.0 | 0.0020 | 28.0 | 0.0048 | ||
5300 | 22.0 | 0.0014 | 53.0 | 0.0045 | 47.0 | 0.0043 | 27.0 | 0.0023 | 27.0 | 0.0042 | ||
5500 | 22.0 | 0.0012 | 52.0 | 0.0041 | 47.0 | 0.0042 | 26.0 | 0.0018 | 26.0 | 0.0036 |
Data | T-LU/ME | ADMM | SPADM | IPADM | PADML | IADML | ||||||
Iter. | T-A | Iter. | T-A | Iter. | T-A | Iter. | T-A | Iter. | T-A | |||
|
T-LU |
300 | 30.7 | 0.5069 | 75.2 | 0.5898 | 66.9 | 0.5286 | 36.3 | 0.2987 | 34.1 | 0.2784 |
T-ME |
500 | 17.9 | 0.3010 | 49.9 | 0.4102 | 42.5 | 0.3555 | 26.5 | 0.2260 | 24.5 | 0.2034 | |
800 | 11.5 | 0.1888 | 36.0 | 0.2854 | 31.7 | 0.2539 | 20.1 | 0.1695 | 18.3 | 0.1507 | ||
1000 | 10.3 | 0.1716 | 31.6 | 0.2537 | 27.3 | 0.2232 | 18.4 | 0.1578 | 17.1 | 0.1437 | ||
T-LU |
300 | 44.9 | 1.4417 | 111.5 | 1.9481 | 95.9 | 1.6807 | 50.2 | 0.9149 | 47.8 | 0.8656 | |
T-ME |
500 | 27.9 | 0.9129 | 73.1 | 1.2882 | 62.1 | 1.0911 | 36.0 | 0.6608 | 33.7 | 0.6198 | |
800 | 17.9 | 0.5888 | 53.2 | 0.9468 | 42.5 | 0.7578 | 27.9 | 0.5178 | 25.4 | 0.4786 | ||
1000 | 14.3 | 0.4787 | 41.1 | 0.7307 | 35.1 | 0.6266 | 24.6 | 0.4576 | 22.9 | 0.4221 | ||
T-LU=118.826 | 500 | 46.8 | 1.7680 | 86.9 | 4.0866 | 67.2 | 3.1514 | 47.2 | 2.2569 | 47.0 | 2.2453 | |
T-ME=16.141 | 1000 | 24.3 | 0.9105 | 51.0 | 2.3636 | 45.0 | 2.0981 | 27.6 | 1.3902 | 26.1 | 1.2590 | |
1500 | 16.1 | 0.6303 | 40.9 | 1.9333 | 36.0 | 1.6943 | 20.7 | 0.9949 | 19.5 | 0.9493 | ||
2000 | 12.1 | 0.4829 | 28.9 | 1.3641 | 25.9 | 1.2279 | 18.8 | 0.9107 | 17.0 | 0.8338 | ||
T-LU=158.214 | 500 | 48.0 | 4.7976 | 134.3 | 8.7370 | 116.7 | 7.6785 | 61.6 | 4.1815 | 59.7 | 3.9039 | |
T-ME=24.249 | 1000 | 23.4 | 2.3597 | 80.7 | 5.3301 | 70.8 | 5.7369 | 41.6 | 3.6014 | 38.1 | 2.6846 | |
1500 | 17.4 | 1.4561 | 55.7 | 3.0283 | 49.0 | 2.6745 | 31.4 | 1.7645 | 29.1 | 1.6197 | ||
2000 | 13.7 | 1.1631 | 47.0 | 2.6077 | 41.2 | 2.2898 | 29.1 | 1.6362 | 26.3 | 1.4693 |
Data | T-LU/ME | ADMM | SPADM | IPADM | PADML | IADML | ||||||
Iter. | T-A | Iter. | T-A | Iter. | T-A | Iter. | T-A | Iter. | T-A | |||
|
T-LU |
300 | 30.7 | 0.5069 | 75.2 | 0.5898 | 66.9 | 0.5286 | 36.3 | 0.2987 | 34.1 | 0.2784 |
T-ME |
500 | 17.9 | 0.3010 | 49.9 | 0.4102 | 42.5 | 0.3555 | 26.5 | 0.2260 | 24.5 | 0.2034 | |
800 | 11.5 | 0.1888 | 36.0 | 0.2854 | 31.7 | 0.2539 | 20.1 | 0.1695 | 18.3 | 0.1507 | ||
1000 | 10.3 | 0.1716 | 31.6 | 0.2537 | 27.3 | 0.2232 | 18.4 | 0.1578 | 17.1 | 0.1437 | ||
T-LU |
300 | 44.9 | 1.4417 | 111.5 | 1.9481 | 95.9 | 1.6807 | 50.2 | 0.9149 | 47.8 | 0.8656 | |
T-ME |
500 | 27.9 | 0.9129 | 73.1 | 1.2882 | 62.1 | 1.0911 | 36.0 | 0.6608 | 33.7 | 0.6198 | |
800 | 17.9 | 0.5888 | 53.2 | 0.9468 | 42.5 | 0.7578 | 27.9 | 0.5178 | 25.4 | 0.4786 | ||
1000 | 14.3 | 0.4787 | 41.1 | 0.7307 | 35.1 | 0.6266 | 24.6 | 0.4576 | 22.9 | 0.4221 | ||
T-LU=118.826 | 500 | 46.8 | 1.7680 | 86.9 | 4.0866 | 67.2 | 3.1514 | 47.2 | 2.2569 | 47.0 | 2.2453 | |
T-ME=16.141 | 1000 | 24.3 | 0.9105 | 51.0 | 2.3636 | 45.0 | 2.0981 | 27.6 | 1.3902 | 26.1 | 1.2590 | |
1500 | 16.1 | 0.6303 | 40.9 | 1.9333 | 36.0 | 1.6943 | 20.7 | 0.9949 | 19.5 | 0.9493 | ||
2000 | 12.1 | 0.4829 | 28.9 | 1.3641 | 25.9 | 1.2279 | 18.8 | 0.9107 | 17.0 | 0.8338 | ||
T-LU=158.214 | 500 | 48.0 | 4.7976 | 134.3 | 8.7370 | 116.7 | 7.6785 | 61.6 | 4.1815 | 59.7 | 3.9039 | |
T-ME=24.249 | 1000 | 23.4 | 2.3597 | 80.7 | 5.3301 | 70.8 | 5.7369 | 41.6 | 3.6014 | 38.1 | 2.6846 | |
1500 | 17.4 | 1.4561 | 55.7 | 3.0283 | 49.0 | 2.6745 | 31.4 | 1.7645 | 29.1 | 1.6197 | ||
2000 | 13.7 | 1.1631 | 47.0 | 2.6077 | 41.2 | 2.2898 | 29.1 | 1.6362 | 26.3 | 1.4693 |
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