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Cooperation in traffic network problems via evolutionary split variational inequalities
ADMM-type methods for generalized multi-facility Weber problem
1. | Department of Mathematics, College of Science, Nanjing University of Aeronautics and Astronautics, 29 Yudao Street, Qinhuai District, Nanjing, 210016, China |
2. | School of Information and Mathematics, Yangtze University, 1 Nanhuan Road, Jingzhou, 434023, China |
3. | Business School, Nankai University, 94 Weijin Road, Nankai District, Tianjin, 300071, China |
The well-known multi-facility Weber problem (MFWP) is one of fundamental models in facility location. With the aim of enhancing the practical applicability of MFWP, this paper considers a generalized multi-facility Weber problem (GMFWP), where the gauge is used to measure distances and the locational constraints are imposed to new facilities. This paper focuses on developing efficient numerical methods based on alternating direction method of multipliers (ADMM) to solve GMFWP. Specifically, GMFWP is equivalently reformulated into a minmax problem with special structure and then some ADMM-type methods are proposed for its primal problem. Global convergence of proposed methods for GMFWP is established under mild assumptions. Preliminary numerical results are reported to verify the effectiveness of proposed methods.
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, Now Foundations and Trends, 2011.
doi: 10.1561/9781601984616. |
[2] |
E. Carrizosa, E. Conde, M. Muñoz-Márquez and J. Puerto,
Simpson points in planar problems with locational constraints. The polyhedral-gauge case, Math. Oper. Res., 22 (1997), 291-300.
doi: 10.1287/moor.22.2.291. |
[3] |
M. Cera, J. A. Mesa, F. A. Ortega and F. Plastria,
Locating a central hunter on the plane, J. Optim. Theory Appl., 136 (2008), 155-166.
doi: 10.1007/s10957-007-9293-y. |
[4] |
C. Chen, B. He, Y. Ye and X. Yuan,
The direct extension of ADMM for multi-block convex minimization problems is not necessarily convergent, Math. Program., 155 (2016), 57-79.
doi: 10.1007/s10107-014-0826-5. |
[5] |
Y.-H. Dai, D. Han, X. Yuan and W. Zhang,
A sequential updating scheme of the Lagrange multiplier for separable convex programming, Math. Comp., 86 (2017), 315-343.
doi: 10.1090/mcom/3104. |
[6] |
F. Daneshzand and R. Shoeleh, Multifacility location problem, in Facility Location, Contributions to Management Science, Physica, Heidelberg, 2009, 69–92.
doi: 10.1007/978-3-7908-2151-2_4. |
[7] |
Z. Drezner, Facility Location. A Survey of Applications and Methods, Springer Series in Operations Research, Springer-Verlag, New York, 1995. |
[8] |
Z. Drezner and H. W. Hamacher, Facility Location. Applications and Theory, Springer-Verlag, Berlin, 2002. |
[9] |
J. W. Eyster, J. A. White and W. W. Wierwille,
On solving multifacility location problems using a hyperboloid approximation procedure, AIIE Trans., 5 (1973), 1-6.
doi: 10.1080/05695557308974875. |
[10] |
M. Fortin and R. Glowinski, On decomposition-coordination methods using an augmented Lagrangian, in Studies in Mathematics and Its Applications, 15, Elsevier, Amsterdam, 1983, 97–146.
doi: 10.1016/S0168-2024(08)70028-6. |
[11] |
D. Gabay and B. Mercier,
A dual algorithm for the solution of nonlinear variational problems via finite element approximation, Comput. Math. Appl., 2 (1976), 17-40.
doi: 10.1016/0898-1221(76)90003-1. |
[12] |
R. Glowinski, Lectures on Numerical Methods for Non-linear Variational Problems, Research Lectures on Mathematics and Physics, 65, Tata Institute of Fundamental Research, Bombay; sh Springer-Verlag, Berlin-New York, 1980. |
[13] |
R. Glowinski and A. Marrocco, 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] |
K. Guo, D. Han and X. Yuan,
Convergence analysis of Douglas–Rachford splitting method for "strongly + weakly" convex programming, SIAM J. Numer. Anal., 55 (2017), 1549-1577.
doi: 10.1137/16M1078604. |
[15] |
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. |
[16] |
B. He, X. Yuan and W. Zhang,
A customized proximal point algorithm for convex minimization with linear constraints, Comput. Optim. Appl., 56 (2013), 559-572.
doi: 10.1007/s10589-013-9564-5. |
[17] |
B. S. He,
A modified projection and contraction method for a class of linear complementarity problems, J. Comput. Math., 14 (1996), 54-63.
|
[18] |
J. Jiang, S. Zhang, Y. Lv, X. Du and Z. Yan,
An ADMM-based location-allocation algorithm for nonconvex constrained multi-source Weber problem under gauge, J. Global Optim., 76 (2020), 793-818.
doi: 10.1007/s10898-019-00796-9. |
[19] |
J. Jiang, S. Zhang, S. Zhang and J. Wen,
A variational inequality approach for constrained multifacility Weber problem under gauge, J. Ind. Manag. Optim., 14 (2018), 1085-1104.
doi: 10.3934/jimo.2017091. |
[20] |
I. N. Katz and S. R. Vogl,
A Weiszfeld algorithm for the solution of an asymmetric extension of the generalized Fermat location problem, Comput. Math. Appl., 59 (2010), 399-410.
doi: 10.1016/j.camwa.2009.07.007. |
[21] |
X. Li, D. Sun and K.-C. Toh,
A Schur complement based semi-proximal ADMM for convex quadratic conic programming and extensions, Math. Program., 155 (2016), 333-373.
doi: 10.1007/s10107-014-0850-5. |
[22] |
Z. Lin, R. Liu and H. Li,
Linearized alternating direction method with parallel splitting and adaptive penalty for separable convex programs in machine learning, Mach. Learn., 99 (2015), 287-325.
doi: 10.1007/s10994-014-5469-5. |
[23] |
R. F. Love and J. G. Morris,
Mathematical models of road travel distances, Manag. Sci., 25 (1979), 117-210.
doi: 10.1287/mnsc.25.2.130. |
[24] |
W. Miehle,
Link-length minimization in networks, Operations Res., 6 (1958), 232-243.
doi: 10.1287/opre.6.2.232. |
[25] |
H. Minkowski, Theorie der Konvexen Körper, Gesammelte Abhandlungen, Teubner, Berlin, 1911. |
[26] |
L. M. Ostresh,
The multifacility location problem: applications and descent theorems, J. Regional. Sci., 17 (2006), 409-419.
doi: 10.1111/j.1467-9787.1977.tb00511.x. |
[27] |
Y. Peng, A. Ganesh, J. Wright, W. Xu and Y. Ma, RASL: Robust alignment by sparse and low-rank decomposition for linearly correlated images, 2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, San Francisco, CA, 2010.
doi: 10.1109/CVPR.2010.5540138. |
[28] |
F. Plastria,
Asymmetric distances, semidirected networks and majority in Fermat-Weber problems, Ann. Oper. Res., 167 (2009), 121-155.
doi: 10.1007/s10479-008-0351-0. |
[29] |
F. Plastria, The Weiszfeld algorithm: Proof, amendments, and extensions, in Foundations of Location Analysis, International Series in Operations Research & Management Science, 155, Springer, New York, 2011, 357–389.
doi: 10.1007/978-1-4419-7572-0_16. |
[30] |
J. B. Rosen and G. L. Xue,
On the convergence of a hyperboloid approximation procedure for the perturbed Euclidean multifacility location problem, Oper. Res., 41 (1993), 1164-1171.
doi: 10.1287/opre.41.6.1164. |
[31] |
J. B. Rosen and G. L. Xue,
On the convergence of Miehle's algorithm for the Euclidean multifacility location problem, Oper. Res., 40 (1992), 188-191.
doi: 10.1287/opre.40.1.188. |
[32] |
D. Sun, K.-C. Toh and L. Yang,
A convergent 3-block semiproximal alternating direction method of multipliers for conic programming with 4-type constraints, SIAM J. Optim., 25 (2015), 882-915.
doi: 10.1137/140964357. |
[33] |
M. Tao and X. Yuan,
Recovering low-rank and sparse components of matrices from incomplete and noisy observations, SIAM J. Optim., 21 (2011), 57-81.
doi: 10.1137/100781894. |
[34] |
X. Wang and X. Yuan, The linearized alternating direction method of multipliers for Dantzig selector, SIAM J. Sci. Comput., 34 (2012), A2792–A2811.
doi: 10.1137/110833543. |
[35] |
J. E. Ward and R. E. Wendell,
Using block norms for location modeling, Oper. Res., 33 (1985), 1074-1090.
doi: 10.1287/opre.33.5.1074. |
[36] |
E. Weiszfeld,
Sur le point pour lequel la somme des distances de $n$ points donnés est minimum, Tohoku Math. J., 43 (1937), 355-386.
|
[37] |
Z. Wen, D. Goldfarb and W. Yin,
Alternating direction augmented Lagrangian methods for semidefinite programming, Math. Program. Comput., 2 (2010), 203-230.
doi: 10.1007/s12532-010-0017-1. |
[38] |
C. Witzgall, Optimal Location of a Central Facility, Mathematical Models and Concepts, Report 8388, National Bureau of Standards, Washington D.C., 1964. |
[39] |
X. Zhang, M. Burger and S. Osher,
A unified primal-dual algorithm framework based on Bregman iteration, J. Sci. Comput., 46 (2011), 20-46.
doi: 10.1007/s10915-010-9408-8. |
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, Now Foundations and Trends, 2011.
doi: 10.1561/9781601984616. |
[2] |
E. Carrizosa, E. Conde, M. Muñoz-Márquez and J. Puerto,
Simpson points in planar problems with locational constraints. The polyhedral-gauge case, Math. Oper. Res., 22 (1997), 291-300.
doi: 10.1287/moor.22.2.291. |
[3] |
M. Cera, J. A. Mesa, F. A. Ortega and F. Plastria,
Locating a central hunter on the plane, J. Optim. Theory Appl., 136 (2008), 155-166.
doi: 10.1007/s10957-007-9293-y. |
[4] |
C. Chen, B. He, Y. Ye and X. Yuan,
The direct extension of ADMM for multi-block convex minimization problems is not necessarily convergent, Math. Program., 155 (2016), 57-79.
doi: 10.1007/s10107-014-0826-5. |
[5] |
Y.-H. Dai, D. Han, X. Yuan and W. Zhang,
A sequential updating scheme of the Lagrange multiplier for separable convex programming, Math. Comp., 86 (2017), 315-343.
doi: 10.1090/mcom/3104. |
[6] |
F. Daneshzand and R. Shoeleh, Multifacility location problem, in Facility Location, Contributions to Management Science, Physica, Heidelberg, 2009, 69–92.
doi: 10.1007/978-3-7908-2151-2_4. |
[7] |
Z. Drezner, Facility Location. A Survey of Applications and Methods, Springer Series in Operations Research, Springer-Verlag, New York, 1995. |
[8] |
Z. Drezner and H. W. Hamacher, Facility Location. Applications and Theory, Springer-Verlag, Berlin, 2002. |
[9] |
J. W. Eyster, J. A. White and W. W. Wierwille,
On solving multifacility location problems using a hyperboloid approximation procedure, AIIE Trans., 5 (1973), 1-6.
doi: 10.1080/05695557308974875. |
[10] |
M. Fortin and R. Glowinski, On decomposition-coordination methods using an augmented Lagrangian, in Studies in Mathematics and Its Applications, 15, Elsevier, Amsterdam, 1983, 97–146.
doi: 10.1016/S0168-2024(08)70028-6. |
[11] |
D. Gabay and B. Mercier,
A dual algorithm for the solution of nonlinear variational problems via finite element approximation, Comput. Math. Appl., 2 (1976), 17-40.
doi: 10.1016/0898-1221(76)90003-1. |
[12] |
R. Glowinski, Lectures on Numerical Methods for Non-linear Variational Problems, Research Lectures on Mathematics and Physics, 65, Tata Institute of Fundamental Research, Bombay; sh Springer-Verlag, Berlin-New York, 1980. |
[13] |
R. Glowinski and A. Marrocco, 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] |
K. Guo, D. Han and X. Yuan,
Convergence analysis of Douglas–Rachford splitting method for "strongly + weakly" convex programming, SIAM J. Numer. Anal., 55 (2017), 1549-1577.
doi: 10.1137/16M1078604. |
[15] |
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. |
[16] |
B. He, X. Yuan and W. Zhang,
A customized proximal point algorithm for convex minimization with linear constraints, Comput. Optim. Appl., 56 (2013), 559-572.
doi: 10.1007/s10589-013-9564-5. |
[17] |
B. S. He,
A modified projection and contraction method for a class of linear complementarity problems, J. Comput. Math., 14 (1996), 54-63.
|
[18] |
J. Jiang, S. Zhang, Y. Lv, X. Du and Z. Yan,
An ADMM-based location-allocation algorithm for nonconvex constrained multi-source Weber problem under gauge, J. Global Optim., 76 (2020), 793-818.
doi: 10.1007/s10898-019-00796-9. |
[19] |
J. Jiang, S. Zhang, S. Zhang and J. Wen,
A variational inequality approach for constrained multifacility Weber problem under gauge, J. Ind. Manag. Optim., 14 (2018), 1085-1104.
doi: 10.3934/jimo.2017091. |
[20] |
I. N. Katz and S. R. Vogl,
A Weiszfeld algorithm for the solution of an asymmetric extension of the generalized Fermat location problem, Comput. Math. Appl., 59 (2010), 399-410.
doi: 10.1016/j.camwa.2009.07.007. |
[21] |
X. Li, D. Sun and K.-C. Toh,
A Schur complement based semi-proximal ADMM for convex quadratic conic programming and extensions, Math. Program., 155 (2016), 333-373.
doi: 10.1007/s10107-014-0850-5. |
[22] |
Z. Lin, R. Liu and H. Li,
Linearized alternating direction method with parallel splitting and adaptive penalty for separable convex programs in machine learning, Mach. Learn., 99 (2015), 287-325.
doi: 10.1007/s10994-014-5469-5. |
[23] |
R. F. Love and J. G. Morris,
Mathematical models of road travel distances, Manag. Sci., 25 (1979), 117-210.
doi: 10.1287/mnsc.25.2.130. |
[24] |
W. Miehle,
Link-length minimization in networks, Operations Res., 6 (1958), 232-243.
doi: 10.1287/opre.6.2.232. |
[25] |
H. Minkowski, Theorie der Konvexen Körper, Gesammelte Abhandlungen, Teubner, Berlin, 1911. |
[26] |
L. M. Ostresh,
The multifacility location problem: applications and descent theorems, J. Regional. Sci., 17 (2006), 409-419.
doi: 10.1111/j.1467-9787.1977.tb00511.x. |
[27] |
Y. Peng, A. Ganesh, J. Wright, W. Xu and Y. Ma, RASL: Robust alignment by sparse and low-rank decomposition for linearly correlated images, 2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, San Francisco, CA, 2010.
doi: 10.1109/CVPR.2010.5540138. |
[28] |
F. Plastria,
Asymmetric distances, semidirected networks and majority in Fermat-Weber problems, Ann. Oper. Res., 167 (2009), 121-155.
doi: 10.1007/s10479-008-0351-0. |
[29] |
F. Plastria, The Weiszfeld algorithm: Proof, amendments, and extensions, in Foundations of Location Analysis, International Series in Operations Research & Management Science, 155, Springer, New York, 2011, 357–389.
doi: 10.1007/978-1-4419-7572-0_16. |
[30] |
J. B. Rosen and G. L. Xue,
On the convergence of a hyperboloid approximation procedure for the perturbed Euclidean multifacility location problem, Oper. Res., 41 (1993), 1164-1171.
doi: 10.1287/opre.41.6.1164. |
[31] |
J. B. Rosen and G. L. Xue,
On the convergence of Miehle's algorithm for the Euclidean multifacility location problem, Oper. Res., 40 (1992), 188-191.
doi: 10.1287/opre.40.1.188. |
[32] |
D. Sun, K.-C. Toh and L. Yang,
A convergent 3-block semiproximal alternating direction method of multipliers for conic programming with 4-type constraints, SIAM J. Optim., 25 (2015), 882-915.
doi: 10.1137/140964357. |
[33] |
M. Tao and X. Yuan,
Recovering low-rank and sparse components of matrices from incomplete and noisy observations, SIAM J. Optim., 21 (2011), 57-81.
doi: 10.1137/100781894. |
[34] |
X. Wang and X. Yuan, The linearized alternating direction method of multipliers for Dantzig selector, SIAM J. Sci. Comput., 34 (2012), A2792–A2811.
doi: 10.1137/110833543. |
[35] |
J. E. Ward and R. E. Wendell,
Using block norms for location modeling, Oper. Res., 33 (1985), 1074-1090.
doi: 10.1287/opre.33.5.1074. |
[36] |
E. Weiszfeld,
Sur le point pour lequel la somme des distances de $n$ points donnés est minimum, Tohoku Math. J., 43 (1937), 355-386.
|
[37] |
Z. Wen, D. Goldfarb and W. Yin,
Alternating direction augmented Lagrangian methods for semidefinite programming, Math. Program. Comput., 2 (2010), 203-230.
doi: 10.1007/s12532-010-0017-1. |
[38] |
C. Witzgall, Optimal Location of a Central Facility, Mathematical Models and Concepts, Report 8388, National Bureau of Standards, Washington D.C., 1964. |
[39] |
X. Zhang, M. Burger and S. Osher,
A unified primal-dual algorithm framework based on Bregman iteration, J. Sci. Comput., 46 (2011), 20-46.
doi: 10.1007/s10915-010-9408-8. |
Problem | MA | GMA | ADMM-a | ADMM-b | ADMM-c | |
E1 | CPU | 0.0003 | 0.0005 | 0.0117 | 0.0096 | 0.0070 |
Iter. | 2 | 9 | 240 | 135 | 62 | |
Obj. | 182.23 | 182.43 | 181.42 | 181.42 | 181.42 | |
E2 | CPU | 0.0069 | 0.0004 | 0.0127 | 0.0097 | 0.0095 |
Iter. | 206 | 5 | 240 | 135 | 82 | |
Obj. | 182.19 | 182.68 | 181.42 | 181.42 | 181.42 | |
E1 |
CPU | / | / | 0.0125 | 0.0084 | 0.0081 |
Iter. | 244.99 | 118.45 | 70.28 | |||
Obj. | 181.42 | 181.42 | 181.42 | |||
E2 |
CPU | / | / | 0.0117 | 0.0076 | 0.0073 |
Iter. | 244.31 | 117.05 | 69.32 | |||
Obj. | 181.42 | 181.42 | 181.42 | |||
E3 | CPU | 0.1774 | 0.1670 | 0.0095 | 0.0109 | 0.0092 |
Iter. | 4502.30 | 56.70 | 203.15 | 168.30 | 89.10 | |
Obj. | 65724.12 | 68142.84 | 65317.20 | 65317.20 | 65317.19 |
Problem | MA | GMA | ADMM-a | ADMM-b | ADMM-c | |
E1 | CPU | 0.0003 | 0.0005 | 0.0117 | 0.0096 | 0.0070 |
Iter. | 2 | 9 | 240 | 135 | 62 | |
Obj. | 182.23 | 182.43 | 181.42 | 181.42 | 181.42 | |
E2 | CPU | 0.0069 | 0.0004 | 0.0127 | 0.0097 | 0.0095 |
Iter. | 206 | 5 | 240 | 135 | 82 | |
Obj. | 182.19 | 182.68 | 181.42 | 181.42 | 181.42 | |
E1 |
CPU | / | / | 0.0125 | 0.0084 | 0.0081 |
Iter. | 244.99 | 118.45 | 70.28 | |||
Obj. | 181.42 | 181.42 | 181.42 | |||
E2 |
CPU | / | / | 0.0117 | 0.0076 | 0.0073 |
Iter. | 244.31 | 117.05 | 69.32 | |||
Obj. | 181.42 | 181.42 | 181.42 | |||
E3 | CPU | 0.1774 | 0.1670 | 0.0095 | 0.0109 | 0.0092 |
Iter. | 4502.30 | 56.70 | 203.15 | 168.30 | 89.10 | |
Obj. | 65724.12 | 68142.84 | 65317.20 | 65317.20 | 65317.19 |
m | HAP | GHAP | SHAP | SGHAP | ADMM-a | ADMM-b | ADMM-c | |||||
2 | CPU | 0.2439 | 0.5834 | 1.4800 | 0.1494 | 0.3407 | 0.8399 | 0.6142 | 0.5791 | 0.0589 | 0.0742 | 0.0556 |
Iter. | 99.65 | 241.65 | 612.35 | 61.05 | 137.75 | 348.45 | 254.45 | 237.15 | 355.18 | 407.11 | 208.99 | |
Dobj. | 25.92 | 8.07 | 2.14 | 25.92 | 8.07 | 2.14 | 2.14 | 2.14 | 0 | -0.06 | -0.43 | |
4 | CPU | 1.3323 | 3.5621 | 9.5611 | 0.7189 | 1.9040 | 4.9394 | 2.0067 | 1.6763 | 0.0783 | 0.0981 | 0.0664 |
Iter. | 265.55 | 709.90 | 1905.55 | 143.55 | 369.95 | 987.00 | 406.90 | 336.35 | 260.72 | 296.51 | 126.30 | |
Dobj. | 91.88 | 28.71 | 8.22 | 91.88 | 28.71 | 8.22 | 8.22 | 8.21 | 0 | -0.10 | -0.66 | |
6 | CPU | 3.3433 | 8.7684 | 23.9782 | 1.7638 | 4.6869 | 12.6427 | 4.2423 | 2.9516 | 0.1061 | 0.1400 | 0.1041 |
Iter. | 420.80 | 1130.30 | 3035.30 | 226.10 | 600.10 | 1615.80 | 439.00 | 370.10 | 243.37 | 279.75 | 121.83 | |
Dobj. | 170.51 | 53.47 | 15.83 | 170.51 | 53.47 | 15.82 | 15.81 | 15.81 | 0 | -0.22 | -1.35 | |
8 | CPU | 6.1408 | 16.4112 | 44.5874 | 3.2534 | 8.7586 | 23.5766 | 4.9028 | 3.8896 | 0.1092 | 0.1859 | 0.1079 |
Iter. | 614.00 | 1666.40 | 4485.80 | 323.00 | 867.80 | 2337.60 | 490.80 | 384.80 | 197.44 | 260.48 | 118.38 | |
Dobj. | 233.83 | 73.59 | 22.29 | 233.83 | 73.59 | 22.29 | 22.30 | 22.28 | 0 | -0.30 | -1.47 | |
10 | CPU | 9.8082 | 27.1206 | 73.7662 | 5.2726 | 14.7574 | 39.6916 | 12.3670 | 7.7926 | 0.1790 | 0.2487 | 0.1668 |
Iter. | 763.62 | 2099.02 | 5717.02 | 407.22 | 1116.22 | 3052.62 | 942.82 | 597.22 | 176.86 | 235.07 | 113.66 | |
Dobj. | 278.14 | 86.02 | 24.69 | 278.14 | 86.02 | 24.68 | 24.69 | 24.67 | 0 | 0.33 | -1.91 |
m | HAP | GHAP | SHAP | SGHAP | ADMM-a | ADMM-b | ADMM-c | |||||
2 | CPU | 0.2439 | 0.5834 | 1.4800 | 0.1494 | 0.3407 | 0.8399 | 0.6142 | 0.5791 | 0.0589 | 0.0742 | 0.0556 |
Iter. | 99.65 | 241.65 | 612.35 | 61.05 | 137.75 | 348.45 | 254.45 | 237.15 | 355.18 | 407.11 | 208.99 | |
Dobj. | 25.92 | 8.07 | 2.14 | 25.92 | 8.07 | 2.14 | 2.14 | 2.14 | 0 | -0.06 | -0.43 | |
4 | CPU | 1.3323 | 3.5621 | 9.5611 | 0.7189 | 1.9040 | 4.9394 | 2.0067 | 1.6763 | 0.0783 | 0.0981 | 0.0664 |
Iter. | 265.55 | 709.90 | 1905.55 | 143.55 | 369.95 | 987.00 | 406.90 | 336.35 | 260.72 | 296.51 | 126.30 | |
Dobj. | 91.88 | 28.71 | 8.22 | 91.88 | 28.71 | 8.22 | 8.22 | 8.21 | 0 | -0.10 | -0.66 | |
6 | CPU | 3.3433 | 8.7684 | 23.9782 | 1.7638 | 4.6869 | 12.6427 | 4.2423 | 2.9516 | 0.1061 | 0.1400 | 0.1041 |
Iter. | 420.80 | 1130.30 | 3035.30 | 226.10 | 600.10 | 1615.80 | 439.00 | 370.10 | 243.37 | 279.75 | 121.83 | |
Dobj. | 170.51 | 53.47 | 15.83 | 170.51 | 53.47 | 15.82 | 15.81 | 15.81 | 0 | -0.22 | -1.35 | |
8 | CPU | 6.1408 | 16.4112 | 44.5874 | 3.2534 | 8.7586 | 23.5766 | 4.9028 | 3.8896 | 0.1092 | 0.1859 | 0.1079 |
Iter. | 614.00 | 1666.40 | 4485.80 | 323.00 | 867.80 | 2337.60 | 490.80 | 384.80 | 197.44 | 260.48 | 118.38 | |
Dobj. | 233.83 | 73.59 | 22.29 | 233.83 | 73.59 | 22.29 | 22.30 | 22.28 | 0 | -0.30 | -1.47 | |
10 | CPU | 9.8082 | 27.1206 | 73.7662 | 5.2726 | 14.7574 | 39.6916 | 12.3670 | 7.7926 | 0.1790 | 0.2487 | 0.1668 |
Iter. | 763.62 | 2099.02 | 5717.02 | 407.22 | 1116.22 | 3052.62 | 942.82 | 597.22 | 176.86 | 235.07 | 113.66 | |
Dobj. | 278.14 | 86.02 | 24.69 | 278.14 | 86.02 | 24.68 | 24.69 | 24.67 | 0 | 0.33 | -1.91 |
n | m | PC method | PPA | Projection | ADMM-a | ADMM-b | ADMM-c | ||||||
CPU | Iter. | CPU | Iter. | CPU | Iter. | CPU | Iter. | CPU | Iter. | CPU | Iter. | ||
50 | 2 | 0.0206 | 34.08 | 0.0077 | 29.10 | 0.0077 | 28.72 | 0.0017 | 24.58 | 0.0031 | 34.52 | 0.0088 | 78.36 |
4 | 0.0414 | 39.46 | 0.0169 | 34.02 | 0.0160 | 32.20 | 0.0044 | 35.10 | 0.0079 | 45.58 | 0.0194 | 83.76 | |
6 | 0.0838 | 54.34 | 0.0287 | 42.24 | 0.0286 | 42.82 | 0.0068 | 43.94 | 0.0139 | 54.18 | 0.0371 | 111.10 | |
8 | 0.1847 | 59.34 | 0.0407 | 44.12 | 0.0414 | 49.22 | 0.0145 | 73.32 | 0.0324 | 87.78 | 0.0574 | 125.04 | |
10 | 0.3244 | 74.12 | 0.0631 | 57.28 | 0.0622 | 61.02 | 0.0189 | 78.34 | 0.0437 | 88.12 | 0.0956 | 157.16 | |
100 | 2 | 0.0388 | 19.00 | 0.0139 | 30.84 | 0.0137 | 30.42 | 0.0024 | 30.72 | 0.0035 | 38.80 | 0.0083 | 64.70 |
4 | 0.0803 | 19.44 | 0.0242 | 31.58 | 0.0236 | 31.04 | 0.0051 | 38.16 | 0.0075 | 41.82 | 0.0153 | 66.00 | |
6 | 0.1244 | 25.56 | 0.0378 | 33.28 | 0.0377 | 32.62 | 0.0083 | 45.58 | 0.0144 | 50.46 | 0.0240 | 66.42 | |
8 | 0.2245 | 26.16 | 0.0559 | 34.78 | 0.0527 | 32.86 | 0.0113 | 46.92 | 0.0217 | 51.34 | 0.0391 | 70.92 | |
10 | 0.2902 | 33.92 | 0.0733 | 37.16 | 0.0706 | 34.54 | 0.0162 | 54.78 | 0.0328 | 57.96 | 0.0532 | 73.00 | |
500 | 2 | 0.0894 | 8.38 | 0.0506 | 25.24 | 0.0408 | 19.90 | 0.0058 | 35.94 | 0.0068 | 36.38 | 0.0051 | 23.10 |
4 | 0.4026 | 10.04 | 0.1197 | 33.14 | 0.0991 | 26.06 | 0.0167 | 50.44 | 0.0204 | 50.90 | 0.0165 | 28.46 | |
6 | 1.1204 | 12.30 | 0.2026 | 39.44 | 0.1651 | 31.36 | 0.0313 | 71.02 | 0.0393 | 71.12 | 0.0301 | 33.04 | |
8 | 3.3505 | 19.32 | 0.3872 | 56.92 | 0.2895 | 42.24 | 0.0410 | 72.04 | 0.0562 | 73.04 | 0.0354 | 35.76 | |
10 | 5.8515 | 20.12 | 0.4112 | 57.90 | 0.3143 | 45.24 | 0.0539 | 73.12 | 0.0734 | 73.22 | 0.0497 | 38.00 | |
1000 | 2 | 0.2795 | 7.36 | 0.1047 | 29.16 | 0.0853 | 22.04 | 0.0113 | 38.20 | 0.0131 | 39.70 | 0.0088 | 19.34 |
4 | 1.5470 | 9.14 | 0.3028 | 42.30 | 0.2478 | 32.06 | 0.0303 | 57.66 | 0.0341 | 58.22 | 0.0238 | 26.34 | |
6 | 5.7046 | 14.26 | 0.4877 | 48.60 | 0.3990 | 37.96 | 0.0483 | 66.78 | 0.0548 | 67.58 | 0.0415 | 30.82 | |
8 | 12.9352 | 15.64 | 0.7822 | 57.38 | 0.6016 | 44.12 | 0.0708 | 76.92 | 0.0895 | 79.40 | 0.0654 | 36.58 | |
10 | 25.0276 | 21.38 | 0.9654 | 58.16 | 0.7617 | 46.14 | 0.0876 | 81.78 | 0.1094 | 81.96 | 0.0810 | 39.50 | |
2000 | 2 | 1.4541 | 7.20 | 0.2724 | 34.60 | 0.2187 | 26.50 | 0.0278 | 44.90 | 0.0287 | 45.70 | 0.0260 | 18.80 |
4 | 10.4432 | 9.08 | 0.8724 | 57.20 | 0.6840 | 44.10 | 0.0602 | 60.70 | 0.0630 | 62.10 | 0.0577 | 25.80 | |
6 | 26.0975 | 13.60 | 1.4767 | 64.80 | 1.1431 | 48.30 | 0.0850 | 69.90 | 0.0949 | 72.60 | 0.0830 | 29.70 | |
8 | 44.1188 | 10.90 | 1.8913 | 67.30 | 1.3707 | 49.30 | 0.1255 | 77.20 | 0.1313 | 79.70 | 0.1213 | 36.20 | |
10 | 69.0254 | 12.20 | 2.8059 | 73.20 | 2.0799 | 53.40 | 0.1518 | 81.90 | 0.1719 | 88.20 | 0.1470 | 39.10 |
n | m | PC method | PPA | Projection | ADMM-a | ADMM-b | ADMM-c | ||||||
CPU | Iter. | CPU | Iter. | CPU | Iter. | CPU | Iter. | CPU | Iter. | CPU | Iter. | ||
50 | 2 | 0.0206 | 34.08 | 0.0077 | 29.10 | 0.0077 | 28.72 | 0.0017 | 24.58 | 0.0031 | 34.52 | 0.0088 | 78.36 |
4 | 0.0414 | 39.46 | 0.0169 | 34.02 | 0.0160 | 32.20 | 0.0044 | 35.10 | 0.0079 | 45.58 | 0.0194 | 83.76 | |
6 | 0.0838 | 54.34 | 0.0287 | 42.24 | 0.0286 | 42.82 | 0.0068 | 43.94 | 0.0139 | 54.18 | 0.0371 | 111.10 | |
8 | 0.1847 | 59.34 | 0.0407 | 44.12 | 0.0414 | 49.22 | 0.0145 | 73.32 | 0.0324 | 87.78 | 0.0574 | 125.04 | |
10 | 0.3244 | 74.12 | 0.0631 | 57.28 | 0.0622 | 61.02 | 0.0189 | 78.34 | 0.0437 | 88.12 | 0.0956 | 157.16 | |
100 | 2 | 0.0388 | 19.00 | 0.0139 | 30.84 | 0.0137 | 30.42 | 0.0024 | 30.72 | 0.0035 | 38.80 | 0.0083 | 64.70 |
4 | 0.0803 | 19.44 | 0.0242 | 31.58 | 0.0236 | 31.04 | 0.0051 | 38.16 | 0.0075 | 41.82 | 0.0153 | 66.00 | |
6 | 0.1244 | 25.56 | 0.0378 | 33.28 | 0.0377 | 32.62 | 0.0083 | 45.58 | 0.0144 | 50.46 | 0.0240 | 66.42 | |
8 | 0.2245 | 26.16 | 0.0559 | 34.78 | 0.0527 | 32.86 | 0.0113 | 46.92 | 0.0217 | 51.34 | 0.0391 | 70.92 | |
10 | 0.2902 | 33.92 | 0.0733 | 37.16 | 0.0706 | 34.54 | 0.0162 | 54.78 | 0.0328 | 57.96 | 0.0532 | 73.00 | |
500 | 2 | 0.0894 | 8.38 | 0.0506 | 25.24 | 0.0408 | 19.90 | 0.0058 | 35.94 | 0.0068 | 36.38 | 0.0051 | 23.10 |
4 | 0.4026 | 10.04 | 0.1197 | 33.14 | 0.0991 | 26.06 | 0.0167 | 50.44 | 0.0204 | 50.90 | 0.0165 | 28.46 | |
6 | 1.1204 | 12.30 | 0.2026 | 39.44 | 0.1651 | 31.36 | 0.0313 | 71.02 | 0.0393 | 71.12 | 0.0301 | 33.04 | |
8 | 3.3505 | 19.32 | 0.3872 | 56.92 | 0.2895 | 42.24 | 0.0410 | 72.04 | 0.0562 | 73.04 | 0.0354 | 35.76 | |
10 | 5.8515 | 20.12 | 0.4112 | 57.90 | 0.3143 | 45.24 | 0.0539 | 73.12 | 0.0734 | 73.22 | 0.0497 | 38.00 | |
1000 | 2 | 0.2795 | 7.36 | 0.1047 | 29.16 | 0.0853 | 22.04 | 0.0113 | 38.20 | 0.0131 | 39.70 | 0.0088 | 19.34 |
4 | 1.5470 | 9.14 | 0.3028 | 42.30 | 0.2478 | 32.06 | 0.0303 | 57.66 | 0.0341 | 58.22 | 0.0238 | 26.34 | |
6 | 5.7046 | 14.26 | 0.4877 | 48.60 | 0.3990 | 37.96 | 0.0483 | 66.78 | 0.0548 | 67.58 | 0.0415 | 30.82 | |
8 | 12.9352 | 15.64 | 0.7822 | 57.38 | 0.6016 | 44.12 | 0.0708 | 76.92 | 0.0895 | 79.40 | 0.0654 | 36.58 | |
10 | 25.0276 | 21.38 | 0.9654 | 58.16 | 0.7617 | 46.14 | 0.0876 | 81.78 | 0.1094 | 81.96 | 0.0810 | 39.50 | |
2000 | 2 | 1.4541 | 7.20 | 0.2724 | 34.60 | 0.2187 | 26.50 | 0.0278 | 44.90 | 0.0287 | 45.70 | 0.0260 | 18.80 |
4 | 10.4432 | 9.08 | 0.8724 | 57.20 | 0.6840 | 44.10 | 0.0602 | 60.70 | 0.0630 | 62.10 | 0.0577 | 25.80 | |
6 | 26.0975 | 13.60 | 1.4767 | 64.80 | 1.1431 | 48.30 | 0.0850 | 69.90 | 0.0949 | 72.60 | 0.0830 | 29.70 | |
8 | 44.1188 | 10.90 | 1.8913 | 67.30 | 1.3707 | 49.30 | 0.1255 | 77.20 | 0.1313 | 79.70 | 0.1213 | 36.20 | |
10 | 69.0254 | 12.20 | 2.8059 | 73.20 | 2.0799 | 53.40 | 0.1518 | 81.90 | 0.1719 | 88.20 | 0.1470 | 39.10 |
n | m | PC method | PPA | Projection | ADMM-a | ADMM-b | ADMM-c | ||||||
CPU | Iter. | CPU | Iter. | CPU | Iter. | CPU | Iter. | CPU | Iter. | CPU | Iter. | ||
50 | 2 | 0.0226 | 26.76 | 0.0125 | 32.02 | 0.0094 | 23.56 | 0.0062 | 69.28 | 0.0072 | 69.74 | 0.0102 | 131.80 |
4 | 0.1120 | 54.80 | 0.0394 | 58.72 | 0.0352 | 51.36 | 0.0112 | 77.88 | 0.0172 | 89.92 | 0.0367 | 146.72 | |
6 | 0.1453 | 66.66 | 0.0650 | 65.42 | 0.0522 | 51.48 | 0.0241 | 103.96 | 0.0373 | 115.60 | 0.0617 | 148.30 | |
8 | 0.2970 | 73.08 | 0.0920 | 67.32 | 0.0744 | 54.58 | 0.0331 | 109.36 | 0.0574 | 116.42 | 0.0897 | 150.92 | |
10 | 0.4527 | 79.68 | 0.1171 | 67.64 | 0.1045 | 59.74 | 0.0564 | 121.54 | 0.0874 | 122.90 | 0.1414 | 159.46 | |
100 | 2 | 0.0459 | 21.94 | 0.0241 | 34.18 | 0.0182 | 28.14 | 0.0079 | 85.28 | 0.0099 | 89.76 | 0.0109 | 60.30 |
4 | 0.1011 | 27.74 | 0.0426 | 38.96 | 0.0357 | 28.28 | 0.0164 | 90.60 | 0.0206 | 94.42 | 0.0201 | 67.06 | |
6 | 0.2569 | 30.30 | 0.0852 | 47.28 | 0.0757 | 39.06 | 0.0323 | 111.66 | 0.0427 | 113.36 | 0.0455 | 97.42 | |
8 | 0.3799 | 40.12 | 0.1242 | 51.00 | 0.0974 | 40.60 | 0.0491 | 132.14 | 0.0706 | 133.12 | 0.0680 | 98.44 | |
10 | 0.4327 | 42.06 | 0.1516 | 52.70 | 0.1249 | 42.26 | 0.0695 | 138.94 | 0.1077 | 139.06 | 0.0905 | 100.70 | |
500 | 2 | 0.1086 | 8.30 | 0.0868 | 31.66 | 0.0695 | 26.06 | 0.0206 | 95.26 | 0.0218 | 96.42 | 0.0122 | 40.12 |
4 | 0.3752 | 8.46 | 0.1778 | 32.08 | 0.1339 | 27.52 | 0.0437 | 100.30 | 0.0509 | 102.48 | 0.0309 | 40.48 | |
6 | 0.9676 | 9.86 | 0.3297 | 40.86 | 0.2528 | 31.32 | 0.0780 | 114.80 | 0.0915 | 117.78 | 0.0534 | 45.54 | |
8 | 2.6573 | 14.38 | 0.5367 | 49.78 | 0.3985 | 35.96 | 0.1230 | 124.48 | 0.1375 | 127.12 | 0.0855 | 53.44 | |
10 | 5.7330 | 19.54 | 0.7646 | 55.32 | 0.5923 | 42.42 | 0.1628 | 134.62 | 0.1987 | 135.96 | 0.1489 | 69.00 | |
1000 | 2 | 0.3517 | 8.28 | 0.1729 | 32.26 | 0.1312 | 23.92 | 0.0303 | 75.32 | 0.0344 | 80.06 | 0.0202 | 33.08 |
4 | 2.1254 | 11.90 | 0.5449 | 47.58 | 0.3915 | 36.60 | 0.0753 | 100.12 | 0.0809 | 100.38 | 0.0487 | 40.56 | |
6 | 9.3411 | 22.54 | 0.8087 | 50.74 | 0.6333 | 39.40 | 0.1279 | 114.24 | 0.1378 | 115.46 | 0.0909 | 49.84 | |
8 | 20.3124 | 23.86 | 1.2310 | 55.32 | 0.9067 | 41.00 | 0.1770 | 120.34 | 0.1971 | 120.72 | 0.1236 | 50.38 | |
10 | 20.6695 | 25.24 | 1.6418 | 59.94 | 1.1654 | 42.04 | 0.2456 | 130.70 | 0.2699 | 130.74 | 0.1759 | 53.84 | |
2000 | 2 | 1.9512 | 9.20 | 0.3199 | 25.10 | 0.2638 | 19.80 | 0.0647 | 61.30 | 0.0595 | 67.30 | 0.0443 | 32.20 |
4 | 8.0722 | 9.30 | 1.1069 | 45.60 | 0.8127 | 31.20 | 0.1408 | 87.10 | 0.1480 | 91.40 | 0.0816 | 37.00 | |
6 | 34.8119 | 17.70 | 2.7923 | 73.50 | 1.9660 | 52.80 | 0.2685 | 121.90 | 0.2824 | 122.70 | 0.1948 | 57.90 | |
8 | 79.7269 | 19.10 | 3.0159 | 61.50 | 2.5028 | 51.00 | 0.3592 | 126.90 | 0.4011 | 127.70 | 0.2539 | 59.00 | |
10 | 114.2177 | 19.40 | 4.4809 | 71.50 | 3.2529 | 67.70 | 0.5160 | 163.30 | 0.5360 | 170.80 | 0.3418 | 67.50 |
n | m | PC method | PPA | Projection | ADMM-a | ADMM-b | ADMM-c | ||||||
CPU | Iter. | CPU | Iter. | CPU | Iter. | CPU | Iter. | CPU | Iter. | CPU | Iter. | ||
50 | 2 | 0.0226 | 26.76 | 0.0125 | 32.02 | 0.0094 | 23.56 | 0.0062 | 69.28 | 0.0072 | 69.74 | 0.0102 | 131.80 |
4 | 0.1120 | 54.80 | 0.0394 | 58.72 | 0.0352 | 51.36 | 0.0112 | 77.88 | 0.0172 | 89.92 | 0.0367 | 146.72 | |
6 | 0.1453 | 66.66 | 0.0650 | 65.42 | 0.0522 | 51.48 | 0.0241 | 103.96 | 0.0373 | 115.60 | 0.0617 | 148.30 | |
8 | 0.2970 | 73.08 | 0.0920 | 67.32 | 0.0744 | 54.58 | 0.0331 | 109.36 | 0.0574 | 116.42 | 0.0897 | 150.92 | |
10 | 0.4527 | 79.68 | 0.1171 | 67.64 | 0.1045 | 59.74 | 0.0564 | 121.54 | 0.0874 | 122.90 | 0.1414 | 159.46 | |
100 | 2 | 0.0459 | 21.94 | 0.0241 | 34.18 | 0.0182 | 28.14 | 0.0079 | 85.28 | 0.0099 | 89.76 | 0.0109 | 60.30 |
4 | 0.1011 | 27.74 | 0.0426 | 38.96 | 0.0357 | 28.28 | 0.0164 | 90.60 | 0.0206 | 94.42 | 0.0201 | 67.06 | |
6 | 0.2569 | 30.30 | 0.0852 | 47.28 | 0.0757 | 39.06 | 0.0323 | 111.66 | 0.0427 | 113.36 | 0.0455 | 97.42 | |
8 | 0.3799 | 40.12 | 0.1242 | 51.00 | 0.0974 | 40.60 | 0.0491 | 132.14 | 0.0706 | 133.12 | 0.0680 | 98.44 | |
10 | 0.4327 | 42.06 | 0.1516 | 52.70 | 0.1249 | 42.26 | 0.0695 | 138.94 | 0.1077 | 139.06 | 0.0905 | 100.70 | |
500 | 2 | 0.1086 | 8.30 | 0.0868 | 31.66 | 0.0695 | 26.06 | 0.0206 | 95.26 | 0.0218 | 96.42 | 0.0122 | 40.12 |
4 | 0.3752 | 8.46 | 0.1778 | 32.08 | 0.1339 | 27.52 | 0.0437 | 100.30 | 0.0509 | 102.48 | 0.0309 | 40.48 | |
6 | 0.9676 | 9.86 | 0.3297 | 40.86 | 0.2528 | 31.32 | 0.0780 | 114.80 | 0.0915 | 117.78 | 0.0534 | 45.54 | |
8 | 2.6573 | 14.38 | 0.5367 | 49.78 | 0.3985 | 35.96 | 0.1230 | 124.48 | 0.1375 | 127.12 | 0.0855 | 53.44 | |
10 | 5.7330 | 19.54 | 0.7646 | 55.32 | 0.5923 | 42.42 | 0.1628 | 134.62 | 0.1987 | 135.96 | 0.1489 | 69.00 | |
1000 | 2 | 0.3517 | 8.28 | 0.1729 | 32.26 | 0.1312 | 23.92 | 0.0303 | 75.32 | 0.0344 | 80.06 | 0.0202 | 33.08 |
4 | 2.1254 | 11.90 | 0.5449 | 47.58 | 0.3915 | 36.60 | 0.0753 | 100.12 | 0.0809 | 100.38 | 0.0487 | 40.56 | |
6 | 9.3411 | 22.54 | 0.8087 | 50.74 | 0.6333 | 39.40 | 0.1279 | 114.24 | 0.1378 | 115.46 | 0.0909 | 49.84 | |
8 | 20.3124 | 23.86 | 1.2310 | 55.32 | 0.9067 | 41.00 | 0.1770 | 120.34 | 0.1971 | 120.72 | 0.1236 | 50.38 | |
10 | 20.6695 | 25.24 | 1.6418 | 59.94 | 1.1654 | 42.04 | 0.2456 | 130.70 | 0.2699 | 130.74 | 0.1759 | 53.84 | |
2000 | 2 | 1.9512 | 9.20 | 0.3199 | 25.10 | 0.2638 | 19.80 | 0.0647 | 61.30 | 0.0595 | 67.30 | 0.0443 | 32.20 |
4 | 8.0722 | 9.30 | 1.1069 | 45.60 | 0.8127 | 31.20 | 0.1408 | 87.10 | 0.1480 | 91.40 | 0.0816 | 37.00 | |
6 | 34.8119 | 17.70 | 2.7923 | 73.50 | 1.9660 | 52.80 | 0.2685 | 121.90 | 0.2824 | 122.70 | 0.1948 | 57.90 | |
8 | 79.7269 | 19.10 | 3.0159 | 61.50 | 2.5028 | 51.00 | 0.3592 | 126.90 | 0.4011 | 127.70 | 0.2539 | 59.00 | |
10 | 114.2177 | 19.40 | 4.4809 | 71.50 | 3.2529 | 67.70 | 0.5160 | 163.30 | 0.5360 | 170.80 | 0.3418 | 67.50 |
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