
-
Previous Article
Preconditioned inexact Newton-like method for large nonsymmetric eigenvalue problems
- NACO Home
- This Issue
-
Next Article
Direct method to solve linear-quadratic optimal control problems
An alternate gradient method for optimization problems with orthogonality constraints
1. | School of Science, Hebei University of Technology, Tianjin, 300401, P. R. China |
2. | Institute of Mathematics, Hebei University of Technology, Tianjin, 300401, P. R. China |
In this paper, we propose a new alternate gradient (AG) method to solve a class of optimization problems with orthogonal constraints. In particular, our AG method alternately takes several gradient reflection steps followed by one gradient projection step. It is proved that any accumulation point of the iterations generated by the AG method satisfies the first-order optimal condition. Numerical experiments show that our method is efficient.
References:
[1] |
P. A. Absil, R. Mahony and R. Sepulchre, Optimization Algorithms on Matrix Manifolds, Princeton University Press, Princeton, 2008.
doi: 10.1515/9781400830244.![]() ![]() ![]() |
[2] |
T. Abrudan, J. Eriksson and V. Koivunen,
Steepest descent algorithms for optimization under unitary matrix constraint, IEEE Transactions on Signal Processing, 56 (2008), 1134-1147.
doi: 10.1109/TSP.2007.908999. |
[3] |
J. Barzilai and J. M. Borwein,
Two-point step size gradient methods, IMA Journal of Numerical Analysis, 8 (1988), 141-148.
doi: 10.1093/imanum/8.1.141. |
[4] |
A. D'Aspremont, L. E. Ghaoui and J. G. R. G. Lanckriet,
A direct formulation for sparse PCA using semidefinite programming, SIAM Review, 49 (2007), 434-448.
doi: 10.1137/050645506. |
[5] |
Y. Dai and R. Fletcher,
Projected Barzilai-Borwein methods for large-scale box-constrained quadratic programming, Numerische Mathematik, 100 (2005), 21-47.
doi: 10.1007/s00211-004-0569-y. |
[6] |
L. Eldén and H. Park,
A procrustes problem on the Stiefel manifold, Numerische Mathematik, 82 (1999), 599-619.
doi: 10.1007/s002110050432. |
[7] |
A. Edelman, T. A. Arias and S. T. Smith,
The geometry of algorithms with orthogonality constraints, SIAM Journal on Matrix Analysis and Applications, 20 (1998), 303-353.
doi: 10.1137/S0895479895290954. |
[8] |
B. Gao, X. Liu, X. Chen and Y. Yuan,
A new first-order algorithmic framework for optimization problems with orthogonality constraints, SIAM Journal on Optimization, 28 (2018), 302-332.
doi: 10.1137/16M1098759. |
[9] |
J. Hu, B. Jiang and X. Liu et al,
A note on semidefinite programming relaxations for polynomial optimization over a single sphere, Science China Mathematics, 59 (2016), 1543-1560.
doi: 10.1007/s11425-016-0301-5. |
[10] |
B. Jiang and Y. Dai,
A framework of constraint preserving update schemes for optimization on Stiefel manifold, Mathematical Programming, 153 (2015), 535-575.
doi: 10.1007/s10107-014-0816-7. |
[11] |
X. Liu, C. Hao and M. Cheng, A sequential subspace projection method for linear symmetric eigenvalue problem, Asia-Pacific Journal of Operational Research, 30 (2013), 1340003.1-1340003.17.
doi: 10.1142/S0217595913400034. |
[12] |
X. Liu, X. Wang and Z. Wen et al,
On the convergence of the self-consistent field iteration in Kohn-Sham density functional theory, SIAM Journal on Matrix Analysis and Applications, 35 (2014), 546-558.
doi: 10.1137/130911032. |
[13] |
Y. Nishimori and S. Akaho,
Learning algorithms utilizing quasi-geodesic flows on the Stiefel manifold, Neurocomputing, 67 (2005), 106-135.
|
[14] |
A. Sameh and Z. Tong,
The trace minimization method for the symmetric generalized eigenvalue problem, Journal of Computational and Applied Mathematics, 123 (2000), 155-175.
doi: 10.1016/S0377-0427(00)00391-5. |
[15] |
M. Ulbrich, Z. Wen and C. Yang et al, A proximal gradient method for ensemble density functional theory, SIAM Journal on Scientific Computing, 37 (2015), A1975–A2002.
doi: 10.1137/14098973X. |
[16] |
Z. Wen and W. Yin,
A feasible method for optimization with orthogonality constraints, Mathematical Programming, 142 (2013), 397-434.
doi: 10.1007/s10107-012-0584-1. |
[17] |
C. Yang, J. C. Meza and L. Wang,
A constrained optimization algorithm for total energy minimization in electronic structure calculations, Journal of Computational Physics, 217 (2006), 709-721.
doi: 10.1016/j.jcp.2006.01.030. |
[18] |
C. Yang, J. C. Meza and B. Lee et al,
KSSOLV-a MATLAB toolbox for solving the Kohn-Sham equations, ACM Transactions on Mathematical Software, 36 (2009), 1-35.
doi: 10.1145/1499096.1499099. |
[19] |
H. Zhang and W. W. Hager,
A nonmonotone line search technique and its application to unconstrained optimization, SIAM Journal on Optimization, 14 (2004), 1043-1056.
doi: 10.1137/S1052623403428208. |
[20] |
H. Zou, T. Hastie and R. Tibshirani,
Sparse principal component analysis, Journal of Computational and Graphical Statistics, 15 (2006), 265-286.
doi: 10.1198/106186006X113430. |
[21] |
X. Zhang, J. Zhu and Z. Wen, Gradient type optimization methods for electronic structure calculations, SIAM Journal on Scientific Computing, 36 (2014), A265–A289.
doi: 10.1137/130932934. |
show all references
References:
[1] |
P. A. Absil, R. Mahony and R. Sepulchre, Optimization Algorithms on Matrix Manifolds, Princeton University Press, Princeton, 2008.
doi: 10.1515/9781400830244.![]() ![]() ![]() |
[2] |
T. Abrudan, J. Eriksson and V. Koivunen,
Steepest descent algorithms for optimization under unitary matrix constraint, IEEE Transactions on Signal Processing, 56 (2008), 1134-1147.
doi: 10.1109/TSP.2007.908999. |
[3] |
J. Barzilai and J. M. Borwein,
Two-point step size gradient methods, IMA Journal of Numerical Analysis, 8 (1988), 141-148.
doi: 10.1093/imanum/8.1.141. |
[4] |
A. D'Aspremont, L. E. Ghaoui and J. G. R. G. Lanckriet,
A direct formulation for sparse PCA using semidefinite programming, SIAM Review, 49 (2007), 434-448.
doi: 10.1137/050645506. |
[5] |
Y. Dai and R. Fletcher,
Projected Barzilai-Borwein methods for large-scale box-constrained quadratic programming, Numerische Mathematik, 100 (2005), 21-47.
doi: 10.1007/s00211-004-0569-y. |
[6] |
L. Eldén and H. Park,
A procrustes problem on the Stiefel manifold, Numerische Mathematik, 82 (1999), 599-619.
doi: 10.1007/s002110050432. |
[7] |
A. Edelman, T. A. Arias and S. T. Smith,
The geometry of algorithms with orthogonality constraints, SIAM Journal on Matrix Analysis and Applications, 20 (1998), 303-353.
doi: 10.1137/S0895479895290954. |
[8] |
B. Gao, X. Liu, X. Chen and Y. Yuan,
A new first-order algorithmic framework for optimization problems with orthogonality constraints, SIAM Journal on Optimization, 28 (2018), 302-332.
doi: 10.1137/16M1098759. |
[9] |
J. Hu, B. Jiang and X. Liu et al,
A note on semidefinite programming relaxations for polynomial optimization over a single sphere, Science China Mathematics, 59 (2016), 1543-1560.
doi: 10.1007/s11425-016-0301-5. |
[10] |
B. Jiang and Y. Dai,
A framework of constraint preserving update schemes for optimization on Stiefel manifold, Mathematical Programming, 153 (2015), 535-575.
doi: 10.1007/s10107-014-0816-7. |
[11] |
X. Liu, C. Hao and M. Cheng, A sequential subspace projection method for linear symmetric eigenvalue problem, Asia-Pacific Journal of Operational Research, 30 (2013), 1340003.1-1340003.17.
doi: 10.1142/S0217595913400034. |
[12] |
X. Liu, X. Wang and Z. Wen et al,
On the convergence of the self-consistent field iteration in Kohn-Sham density functional theory, SIAM Journal on Matrix Analysis and Applications, 35 (2014), 546-558.
doi: 10.1137/130911032. |
[13] |
Y. Nishimori and S. Akaho,
Learning algorithms utilizing quasi-geodesic flows on the Stiefel manifold, Neurocomputing, 67 (2005), 106-135.
|
[14] |
A. Sameh and Z. Tong,
The trace minimization method for the symmetric generalized eigenvalue problem, Journal of Computational and Applied Mathematics, 123 (2000), 155-175.
doi: 10.1016/S0377-0427(00)00391-5. |
[15] |
M. Ulbrich, Z. Wen and C. Yang et al, A proximal gradient method for ensemble density functional theory, SIAM Journal on Scientific Computing, 37 (2015), A1975–A2002.
doi: 10.1137/14098973X. |
[16] |
Z. Wen and W. Yin,
A feasible method for optimization with orthogonality constraints, Mathematical Programming, 142 (2013), 397-434.
doi: 10.1007/s10107-012-0584-1. |
[17] |
C. Yang, J. C. Meza and L. Wang,
A constrained optimization algorithm for total energy minimization in electronic structure calculations, Journal of Computational Physics, 217 (2006), 709-721.
doi: 10.1016/j.jcp.2006.01.030. |
[18] |
C. Yang, J. C. Meza and B. Lee et al,
KSSOLV-a MATLAB toolbox for solving the Kohn-Sham equations, ACM Transactions on Mathematical Software, 36 (2009), 1-35.
doi: 10.1145/1499096.1499099. |
[19] |
H. Zhang and W. W. Hager,
A nonmonotone line search technique and its application to unconstrained optimization, SIAM Journal on Optimization, 14 (2004), 1043-1056.
doi: 10.1137/S1052623403428208. |
[20] |
H. Zou, T. Hastie and R. Tibshirani,
Sparse principal component analysis, Journal of Computational and Graphical Statistics, 15 (2006), 265-286.
doi: 10.1198/106186006X113430. |
[21] |
X. Zhang, J. Zhu and Z. Wen, Gradient type optimization methods for electronic structure calculations, SIAM Journal on Scientific Computing, 36 (2014), A265–A289.
doi: 10.1137/130932934. |

cpu | iter | fun | KKT violation | feasibility | |
GP | 3.74 | 682.0 | -91.651414 | 3.8477E-04 | 4.0577E-15 |
GR | 2.40 | 502.4 | -91.646891 | 3.8305E-04 | 1.9524E-13 |
AG |
2.01 | 433.4 | -91.646891 | 3.8447E-04 | 4.5582E-15 |
AG |
2.21 | 483.4 | -91.642235 | 3.8370E-04 | 6.2797E-15 |
AG |
2.34 | 503.0 | -91.646891 | 3.8142E-04 | 6.1260E-15 |
cpu | iter | fun | KKT violation | feasibility | |
GP | 3.74 | 682.0 | -91.651414 | 3.8477E-04 | 4.0577E-15 |
GR | 2.40 | 502.4 | -91.646891 | 3.8305E-04 | 1.9524E-13 |
AG |
2.01 | 433.4 | -91.646891 | 3.8447E-04 | 4.5582E-15 |
AG |
2.21 | 483.4 | -91.642235 | 3.8370E-04 | 6.2797E-15 |
AG |
2.34 | 503.0 | -91.646891 | 3.8142E-04 | 6.1260E-15 |
cpu | iter | fun | KKT violation | feasibility | |
GBB | 9.474 | 734.2 | -131.5233622 | 2.6304E-04 | 1.7197E-15 |
AFBB | 13.388 | 326.3 | -131.5249367 | 1.7790E-03 | 2.2349E-15 |
GP | 10.992 | 939.4 | -131.5225679 | 5.4438E-04 | 4.3864E-15 |
GR | 9.764 | 832.6 | -131.5214683 | 5.4198E-04 | 1.8122E-13 |
AG | 8.882 | 749.9 | -131.5221231 | 5.4445E-04 | 6.1051E-15 |
GBB | 11.263 | 780.5 | -218.5897386 | 4.1310E-04 | 1.9883E-15 |
AFBB | 18.575 | 527.3 | -218.5916071 | 3.2611E-03 | 2.9976E-15 |
GP | 15.173 | 1075.2 | -218.5856256 | 7.0932E-04 | 6.6001E-15 |
GR | 11.220 | 820.6 | -218.594843 | 7.0782E-04 | 2.3583E-13 |
AG | 9.986 | 706.5 | -218.5926144 | 7.0900E-04 | 8.0173E-15 |
GBB | 14.344 | 825.5 | -478.6533636 | 6.3350E-04 | 1.8762E-15 |
AFBB | 22.275 | 465.6 | -478.6490255 | 9.4387E-03 | 3.4080E-15 |
GP | 23.287 | 1251.6 | -478.6547359 | 1.1447E-03 | 1.2903E-14 |
GR | 15.833 | 915.2 | -478.6564912 | 1.1402E-03 | 6.0783E-13 |
AG | 15.426 | 872.9 | -478.6534441 | 1.1421E-03 | 1.6879E-14 |
GBB | 31.84 | 1226.0 | -1245.746827 | 1.5488E-03 | 2.1434E-15 |
AFBB | 48.68 | 661.4 | -1245.748650 | 1.8988E-02 | 5.0980E-15 |
GP | 21.86 | 784.0 | -1245.750942 | 8.4329E-05 | 1.5679E-14 |
GR | 17.06 | 656.8 | -1245.750942 | 7.1151E-05 | 8.0287E-13 |
AG | 16.24 | 602.0 | -1245.750942 | 6.7208E-05 | 1.5920E-14 |
GBB | 61.73 | 2142.0 | -5639.853266 | 9.3744E-03 | 2.7954E-15 |
AFBB | 87.39 | 853.6 | -5639.770839 | 1.2351E-01 | 5.1548E-15 |
GP | 29.81 | 861.2 | -5639.853800 | 7.2273E-05 | 1.9845E-14 |
GR | 24.26 | 693.2 | -5639.853800 | 5.9084E-05 | 7.8415E-13 |
AG | 21.82 | 654.0 | -5639.853800 | 5.7129E-05 | 1.9641E-14 |
GBB | 160.670 | 4324.2 | -32893.22537 | 1.4305E-01 | 2.4661E-15 |
AFBB | 195.783 | 902.0 | -32892.54444 | 6.1956E-01 | 7.0825E-15 |
GP | 10.488 | 234.8 | -32893.2292 | 1.3979E-01 | 2.3702E-14 |
GR | 10.261 | 253.2 | -32893.23767 | 1.3921E-01 | 7.1360E-13 |
AG | 8.226 | 189.8 | -32893.23383 | 1.3910E-01 | 1.6358E-13 |
GBB | 181.231 | 4479.6 | -201185.9576 | 5.2503E-01 | 2.5522E-15 |
AFBB | 220.369 | 889.4 | -201182.0639 | 4.6276E+00 | 7.5163E-15 |
GP | 3.832 | 71.0 | -201185.8551 | 8.6024E-01 | 2.5309E-14 |
GR | 5.578 | 122.2 | -201186.0137 | 7.7927E-01 | 6.1877E-13 |
AG | 3.139 | 63.5 | -201185.8558 | 8.6000E-01 | 1.6652E-13 |
cpu | iter | fun | KKT violation | feasibility | |
GBB | 9.474 | 734.2 | -131.5233622 | 2.6304E-04 | 1.7197E-15 |
AFBB | 13.388 | 326.3 | -131.5249367 | 1.7790E-03 | 2.2349E-15 |
GP | 10.992 | 939.4 | -131.5225679 | 5.4438E-04 | 4.3864E-15 |
GR | 9.764 | 832.6 | -131.5214683 | 5.4198E-04 | 1.8122E-13 |
AG | 8.882 | 749.9 | -131.5221231 | 5.4445E-04 | 6.1051E-15 |
GBB | 11.263 | 780.5 | -218.5897386 | 4.1310E-04 | 1.9883E-15 |
AFBB | 18.575 | 527.3 | -218.5916071 | 3.2611E-03 | 2.9976E-15 |
GP | 15.173 | 1075.2 | -218.5856256 | 7.0932E-04 | 6.6001E-15 |
GR | 11.220 | 820.6 | -218.594843 | 7.0782E-04 | 2.3583E-13 |
AG | 9.986 | 706.5 | -218.5926144 | 7.0900E-04 | 8.0173E-15 |
GBB | 14.344 | 825.5 | -478.6533636 | 6.3350E-04 | 1.8762E-15 |
AFBB | 22.275 | 465.6 | -478.6490255 | 9.4387E-03 | 3.4080E-15 |
GP | 23.287 | 1251.6 | -478.6547359 | 1.1447E-03 | 1.2903E-14 |
GR | 15.833 | 915.2 | -478.6564912 | 1.1402E-03 | 6.0783E-13 |
AG | 15.426 | 872.9 | -478.6534441 | 1.1421E-03 | 1.6879E-14 |
GBB | 31.84 | 1226.0 | -1245.746827 | 1.5488E-03 | 2.1434E-15 |
AFBB | 48.68 | 661.4 | -1245.748650 | 1.8988E-02 | 5.0980E-15 |
GP | 21.86 | 784.0 | -1245.750942 | 8.4329E-05 | 1.5679E-14 |
GR | 17.06 | 656.8 | -1245.750942 | 7.1151E-05 | 8.0287E-13 |
AG | 16.24 | 602.0 | -1245.750942 | 6.7208E-05 | 1.5920E-14 |
GBB | 61.73 | 2142.0 | -5639.853266 | 9.3744E-03 | 2.7954E-15 |
AFBB | 87.39 | 853.6 | -5639.770839 | 1.2351E-01 | 5.1548E-15 |
GP | 29.81 | 861.2 | -5639.853800 | 7.2273E-05 | 1.9845E-14 |
GR | 24.26 | 693.2 | -5639.853800 | 5.9084E-05 | 7.8415E-13 |
AG | 21.82 | 654.0 | -5639.853800 | 5.7129E-05 | 1.9641E-14 |
GBB | 160.670 | 4324.2 | -32893.22537 | 1.4305E-01 | 2.4661E-15 |
AFBB | 195.783 | 902.0 | -32892.54444 | 6.1956E-01 | 7.0825E-15 |
GP | 10.488 | 234.8 | -32893.2292 | 1.3979E-01 | 2.3702E-14 |
GR | 10.261 | 253.2 | -32893.23767 | 1.3921E-01 | 7.1360E-13 |
AG | 8.226 | 189.8 | -32893.23383 | 1.3910E-01 | 1.6358E-13 |
GBB | 181.231 | 4479.6 | -201185.9576 | 5.2503E-01 | 2.5522E-15 |
AFBB | 220.369 | 889.4 | -201182.0639 | 4.6276E+00 | 7.5163E-15 |
GP | 3.832 | 71.0 | -201185.8551 | 8.6024E-01 | 2.5309E-14 |
GR | 5.578 | 122.2 | -201186.0137 | 7.7927E-01 | 6.1877E-13 |
AG | 3.139 | 63.5 | -201185.8558 | 8.6000E-01 | 1.6652E-13 |
cpu | iter | fun | KKT violation | feasibility | |
GBB | 2.350 | 193.7 | -162.4333197 | 5.1684E-04 | 1.5849E-15 |
AFBB | 4.323 | 168.4 | -162.3569464 | 8.9896E-02 | 1.0327E-14 |
GP | 9.625 | 833.8 | -162.4646836 | 6.9011E-04 | 4.5461E-15 |
GR | 7.432 | 653.8 | -162.4753778 | 6.8193E-04 | 2.0315E-13 |
AG | 6.443 | 572.0 | -162.4788103 | 6.8824E-04 | 5.2647E-15 |
cpu | iter | fun | KKT violation | feasibility | |
GBB | 2.412 | 212.8 | -162.4445186 | 3.0936E-04 | 1.6845E-15 |
AFBB | 4.108 | 150.2 | -162.3883351 | 9.9274E-02 | 2.3889E-15 |
GP | 10.028 | 892.1 | -162.4445179 | 6.8857E-04 | 4.8457E-15 |
GR | 6.242 | 546.8 | -162.4768871 | 6.8015E-04 | 1.9881E-13 |
AG | 5.447 | 484.1 | -162.4667359 | 6.8719E-04 | 6.6787E-15 |
GBB | 8.38 | 454.0 | -1504.964138 | 2.6761E-03 | 1.8293E-15 |
AFBB | 14.92 | 385.6 | -1504.918549 | 1.6780E-02 | 3.5068E-15 |
GP | 7.20 | 391.8 | -1504.918628 | 1.0776E-04 | 1.2022E-14 |
GR | 5.94 | 337.2 | -1504.918628 | 9.4064E-05 | 6.5906E-13 |
AG | 5.61 | 310.8 | -1504.918628 | 8.8531E-05 | 1.2883E-14 |
GBB | 29.689 | 1243.9 | -8613.088591 | 2.0335E-02 | 2.2557E-15 |
AFBB | 43.338 | 565.2 | -8613.038104 | 1.4066E-01 | 5.1917E-15 |
GP | 10.754 | 401.0 | -8613.086436 | 3.6167E-02 | 1.5379E-14 |
GR | 6.938 | 275.4 | -8613.096615 | 3.6090E-02 | 8.1878E-13 |
AG | 6.811 | 259.9 | -8613.096568 | 3.6147E-02 | 8.1414E-14 |
GBB | 66.861 | 2384.3 | -52514.12943 | 1.0842E-01 | 2.7336E-15 |
AFBB | 85.967 | 657.8 | -52513.74516 | 1.0337E+00 | 5.2309E-15 |
GP | 4.311 | 128.6 | -52514.11926 | 2.2352E-01 | 1.9552E-14 |
GR | 5.342 | 176.4 | -52514.09427 | 2.2260E-01 | 7.8898E-13 |
AG | 3.657 | 114.3 | -52514.08076 | 2.2395E-01 | 1.4507E-13 |
GBB | 101.840 | 2667.2 | -323993.0909 | 5.2023E-01 | 2.4041E-15 |
AFBB | 131.354 | 735.0 | -323989.6253 | 4.9496E+00 | 7.1751E-15 |
GP | 2.297 | 45.4 | -323992.3883 | 1.3801E+00 | 2.1972E-14 |
GR | 2.527 | 59.2 | -323992.4399 | 1.3566E+00 | 8.8005E-13 |
AG | 1.867 | 39.8 | -323992.402 | 1.3720E+00 | 2.3366E-13 |
cpu | iter | fun | KKT violation | feasibility | |
GBB | 2.350 | 193.7 | -162.4333197 | 5.1684E-04 | 1.5849E-15 |
AFBB | 4.323 | 168.4 | -162.3569464 | 8.9896E-02 | 1.0327E-14 |
GP | 9.625 | 833.8 | -162.4646836 | 6.9011E-04 | 4.5461E-15 |
GR | 7.432 | 653.8 | -162.4753778 | 6.8193E-04 | 2.0315E-13 |
AG | 6.443 | 572.0 | -162.4788103 | 6.8824E-04 | 5.2647E-15 |
cpu | iter | fun | KKT violation | feasibility | |
GBB | 2.412 | 212.8 | -162.4445186 | 3.0936E-04 | 1.6845E-15 |
AFBB | 4.108 | 150.2 | -162.3883351 | 9.9274E-02 | 2.3889E-15 |
GP | 10.028 | 892.1 | -162.4445179 | 6.8857E-04 | 4.8457E-15 |
GR | 6.242 | 546.8 | -162.4768871 | 6.8015E-04 | 1.9881E-13 |
AG | 5.447 | 484.1 | -162.4667359 | 6.8719E-04 | 6.6787E-15 |
GBB | 8.38 | 454.0 | -1504.964138 | 2.6761E-03 | 1.8293E-15 |
AFBB | 14.92 | 385.6 | -1504.918549 | 1.6780E-02 | 3.5068E-15 |
GP | 7.20 | 391.8 | -1504.918628 | 1.0776E-04 | 1.2022E-14 |
GR | 5.94 | 337.2 | -1504.918628 | 9.4064E-05 | 6.5906E-13 |
AG | 5.61 | 310.8 | -1504.918628 | 8.8531E-05 | 1.2883E-14 |
GBB | 29.689 | 1243.9 | -8613.088591 | 2.0335E-02 | 2.2557E-15 |
AFBB | 43.338 | 565.2 | -8613.038104 | 1.4066E-01 | 5.1917E-15 |
GP | 10.754 | 401.0 | -8613.086436 | 3.6167E-02 | 1.5379E-14 |
GR | 6.938 | 275.4 | -8613.096615 | 3.6090E-02 | 8.1878E-13 |
AG | 6.811 | 259.9 | -8613.096568 | 3.6147E-02 | 8.1414E-14 |
GBB | 66.861 | 2384.3 | -52514.12943 | 1.0842E-01 | 2.7336E-15 |
AFBB | 85.967 | 657.8 | -52513.74516 | 1.0337E+00 | 5.2309E-15 |
GP | 4.311 | 128.6 | -52514.11926 | 2.2352E-01 | 1.9552E-14 |
GR | 5.342 | 176.4 | -52514.09427 | 2.2260E-01 | 7.8898E-13 |
AG | 3.657 | 114.3 | -52514.08076 | 2.2395E-01 | 1.4507E-13 |
GBB | 101.840 | 2667.2 | -323993.0909 | 5.2023E-01 | 2.4041E-15 |
AFBB | 131.354 | 735.0 | -323989.6253 | 4.9496E+00 | 7.1751E-15 |
GP | 2.297 | 45.4 | -323992.3883 | 1.3801E+00 | 2.1972E-14 |
GR | 2.527 | 59.2 | -323992.4399 | 1.3566E+00 | 8.8005E-13 |
AG | 1.867 | 39.8 | -323992.402 | 1.3720E+00 | 2.3366E-13 |
cpu | iter | fun | KKT violation | feasibility | |
co2, |
|||||
SCF | 31.768 | 17 | -35.124396 | 1.5035E-06 | 6.5360E-15 |
GBB | 32.818 | 53 | -35.124396 | 3.4202E-06 | 9.5236E-14 |
AFBB | 35.728 | 61 | -35.124396 | 1.2646E-06 | 3.5113E-14 |
GP | 32.941 | 46 | -35.124396 | 9.5300E-06 | 3.8009E-15 |
AG | 28.840 | 44 | -35.124396 | 6.0064E-06 | 2.7404E-15 |
c2h6, |
|||||
SCF | 28.294 | 18 | -14.420491 | 1.0304E-06 | 1.5356E-14 |
GBB | 32.388 | 51 | -14.420491 | 9.7967E-06 | 1.0002E-14 |
AFBB | 32.524 | 52 | -14.420491 | 4.7573E-06 | 1.2778E-14 |
GP | 32.879 | 50 | -14.420491 | 9.7401E-06 | 4.2016E-15 |
AG | 31.951 | 50 | -14.420491 | 5.9437E-06 | 4.2785E-15 |
benzene, |
|||||
SCF | 601.321 | 118 | -37.225751 | 1.9756E-06 | 6.9301E-14 |
GBB | 183.137 | 51 | -37.225751 | 7.4389E-06 | 8.7418E-14 |
AFBB | 200.916 | 58 | -37.225751 | 2.0575E-06 | 1.7646E-13 |
GP | 207.944 | 60 | -37.225751 | 9.6828E-06 | 8.3590E-15 |
AG | 200.898 | 56 | -37.225751 | 4.9222E-06 | 9.8340E-15 |
h2o, |
|||||
SCF | 20.570 | 26 | -16.440507 | 9.3011E-07 | 5.5665E-15 |
GBB | 28.780 | 53 | -16.440507 | 9.4006E-06 | 1.8356E-14 |
AFBB | 30.198 | 59 | -16.440507 | 4.8143E-07 | 1.9661E-14 |
GP | 26.768 | 51 | -16.440507 | 6.6455E-06 | 6.5909E-15 |
AG | 23.697 | 44 | -16.440507 | 8.0030E-06 | 4.8574E-14 |
c12h26, |
|||||
SCF | 418.609 | 55 | -81.536092 | 3.9309E-06 | 3.8821E-14 |
GBB | 283.230 | 66 | -81.536092 | 9.6402E-06 | 6.5371E-14 |
AFBB | 269.654 | 61 | -81.536092 | 6.0986E-06 | 1.0709E-13 |
GP | 301.821 | 71 | -81.536092 | 5.9451E-06 | 1.4570E-14 |
AG | 261.392 | 60 | -81.536092 | 6.4962E-06 | 1.7237E-14 |
si2h4, |
|||||
SCF | 36.054 | 19 | -6.300975 | 1.5619E-06 | 8.8784E-15 |
GBB | 43.816 | 70 | -6.300975 | 3.9377E-06 | 3.7868E-14 |
AFBB | 44.630 | 69 | -6.300975 | 4.7737E-06 | 1.5903E-14 |
GP | 32.075 | 53 | -6.300975 | 9.6973E-06 | 3.9623E-14 |
AG | 36.636 | 62 | -6.300975 | 8.4565E-06 | 2.7022E-13 |
nic, |
|||||
SCF | 10.995 | 14 | -23.543530 | 1.2291E-06 | 3.2549E-15 |
GBB | 10.030 | 45 | -23.543530 | 7.4993E-06 | 2.1795E-14 |
AFBB | 9.038 | 45 | -23.543530 | 7.4993E-06 | 1.1159E-14 |
GP | 11.812 | 84 | -23.543530 | 6.9368E-06 | 1.9205E-15 |
AG | 11.162 | 82 | -23.543530 | 6.1287E-06 | 3.3939E-15 |
cpu | iter | fun | KKT violation | feasibility | |
co2, |
|||||
SCF | 31.768 | 17 | -35.124396 | 1.5035E-06 | 6.5360E-15 |
GBB | 32.818 | 53 | -35.124396 | 3.4202E-06 | 9.5236E-14 |
AFBB | 35.728 | 61 | -35.124396 | 1.2646E-06 | 3.5113E-14 |
GP | 32.941 | 46 | -35.124396 | 9.5300E-06 | 3.8009E-15 |
AG | 28.840 | 44 | -35.124396 | 6.0064E-06 | 2.7404E-15 |
c2h6, |
|||||
SCF | 28.294 | 18 | -14.420491 | 1.0304E-06 | 1.5356E-14 |
GBB | 32.388 | 51 | -14.420491 | 9.7967E-06 | 1.0002E-14 |
AFBB | 32.524 | 52 | -14.420491 | 4.7573E-06 | 1.2778E-14 |
GP | 32.879 | 50 | -14.420491 | 9.7401E-06 | 4.2016E-15 |
AG | 31.951 | 50 | -14.420491 | 5.9437E-06 | 4.2785E-15 |
benzene, |
|||||
SCF | 601.321 | 118 | -37.225751 | 1.9756E-06 | 6.9301E-14 |
GBB | 183.137 | 51 | -37.225751 | 7.4389E-06 | 8.7418E-14 |
AFBB | 200.916 | 58 | -37.225751 | 2.0575E-06 | 1.7646E-13 |
GP | 207.944 | 60 | -37.225751 | 9.6828E-06 | 8.3590E-15 |
AG | 200.898 | 56 | -37.225751 | 4.9222E-06 | 9.8340E-15 |
h2o, |
|||||
SCF | 20.570 | 26 | -16.440507 | 9.3011E-07 | 5.5665E-15 |
GBB | 28.780 | 53 | -16.440507 | 9.4006E-06 | 1.8356E-14 |
AFBB | 30.198 | 59 | -16.440507 | 4.8143E-07 | 1.9661E-14 |
GP | 26.768 | 51 | -16.440507 | 6.6455E-06 | 6.5909E-15 |
AG | 23.697 | 44 | -16.440507 | 8.0030E-06 | 4.8574E-14 |
c12h26, |
|||||
SCF | 418.609 | 55 | -81.536092 | 3.9309E-06 | 3.8821E-14 |
GBB | 283.230 | 66 | -81.536092 | 9.6402E-06 | 6.5371E-14 |
AFBB | 269.654 | 61 | -81.536092 | 6.0986E-06 | 1.0709E-13 |
GP | 301.821 | 71 | -81.536092 | 5.9451E-06 | 1.4570E-14 |
AG | 261.392 | 60 | -81.536092 | 6.4962E-06 | 1.7237E-14 |
si2h4, |
|||||
SCF | 36.054 | 19 | -6.300975 | 1.5619E-06 | 8.8784E-15 |
GBB | 43.816 | 70 | -6.300975 | 3.9377E-06 | 3.7868E-14 |
AFBB | 44.630 | 69 | -6.300975 | 4.7737E-06 | 1.5903E-14 |
GP | 32.075 | 53 | -6.300975 | 9.6973E-06 | 3.9623E-14 |
AG | 36.636 | 62 | -6.300975 | 8.4565E-06 | 2.7022E-13 |
nic, |
|||||
SCF | 10.995 | 14 | -23.543530 | 1.2291E-06 | 3.2549E-15 |
GBB | 10.030 | 45 | -23.543530 | 7.4993E-06 | 2.1795E-14 |
AFBB | 9.038 | 45 | -23.543530 | 7.4993E-06 | 1.1159E-14 |
GP | 11.812 | 84 | -23.543530 | 6.9368E-06 | 1.9205E-15 |
AG | 11.162 | 82 | -23.543530 | 6.1287E-06 | 3.3939E-15 |
[1] |
Qinghua Ma, Zuoliang Xu, Liping Wang. Recovery of the local volatility function using regularization and a gradient projection method. Journal of Industrial and Management Optimization, 2015, 11 (2) : 421-437. doi: 10.3934/jimo.2015.11.421 |
[2] |
Gaohang Yu, Shanzhou Niu, Jianhua Ma. Multivariate spectral gradient projection method for nonlinear monotone equations with convex constraints. Journal of Industrial and Management Optimization, 2013, 9 (1) : 117-129. doi: 10.3934/jimo.2013.9.117 |
[3] |
Hanchun Yang, Meimei Zhang, Qin Wang. Global solutions of shock reflection problem for the pressure gradient system. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3387-3428. doi: 10.3934/cpaa.2020150 |
[4] |
Richard A. Norton, David I. McLaren, G. R. W. Quispel, Ari Stern, Antonella Zanna. Projection methods and discrete gradient methods for preserving first integrals of ODEs. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 2079-2098. doi: 10.3934/dcds.2015.35.2079 |
[5] |
Jianjun Zhang, Yunyi Hu, James G. Nagy. A scaled gradient method for digital tomographic image reconstruction. Inverse Problems and Imaging, 2018, 12 (1) : 239-259. doi: 10.3934/ipi.2018010 |
[6] |
José Antonio Carrillo, Yanghong Huang, Francesco Saverio Patacchini, Gershon Wolansky. Numerical study of a particle method for gradient flows. Kinetic and Related Models, 2017, 10 (3) : 613-641. doi: 10.3934/krm.2017025 |
[7] |
Stefan Kindermann. Convergence of the gradient method for ill-posed problems. Inverse Problems and Imaging, 2017, 11 (4) : 703-720. doi: 10.3934/ipi.2017033 |
[8] |
Daniela Saxenhuber, Ronny Ramlau. A gradient-based method for atmospheric tomography. Inverse Problems and Imaging, 2016, 10 (3) : 781-805. doi: 10.3934/ipi.2016021 |
[9] |
Wanyou Cheng, Zixin Chen, Donghui Li. Nomonotone spectral gradient method for sparse recovery. Inverse Problems and Imaging, 2015, 9 (3) : 815-833. doi: 10.3934/ipi.2015.9.815 |
[10] |
Yigui Ou, Yuanwen Liu. A memory gradient method based on the nonmonotone technique. Journal of Industrial and Management Optimization, 2017, 13 (2) : 857-872. doi: 10.3934/jimo.2016050 |
[11] |
Esmail Abdul Fattah, Janet Van Niekerk, Håvard Rue. Smart Gradient - An adaptive technique for improving gradient estimation. Foundations of Data Science, 2022, 4 (1) : 123-136. doi: 10.3934/fods.2021037 |
[12] |
Delio Mugnolo, René Pröpper. Gradient systems on networks. Conference Publications, 2011, 2011 (Special) : 1078-1090. doi: 10.3934/proc.2011.2011.1078 |
[13] |
Yuhong Dai, Ya-xiang Yuan. Analysis of monotone gradient methods. Journal of Industrial and Management Optimization, 2005, 1 (2) : 181-192. doi: 10.3934/jimo.2005.1.181 |
[14] |
Ting Hu. Kernel-based maximum correntropy criterion with gradient descent method. Communications on Pure and Applied Analysis, 2020, 19 (8) : 4159-4177. doi: 10.3934/cpaa.2020186 |
[15] |
C.Y. Wang, M.X. Li. Convergence property of the Fletcher-Reeves conjugate gradient method with errors. Journal of Industrial and Management Optimization, 2005, 1 (2) : 193-200. doi: 10.3934/jimo.2005.1.193 |
[16] |
Yanfei Wang, Dmitry Lukyanenko, Anatoly Yagola. Magnetic parameters inversion method with full tensor gradient data. Inverse Problems and Imaging, 2019, 13 (4) : 745-754. doi: 10.3934/ipi.2019034 |
[17] |
Yu-Ning Yang, Su Zhang. On linear convergence of projected gradient method for a class of affine rank minimization problems. Journal of Industrial and Management Optimization, 2016, 12 (4) : 1507-1519. doi: 10.3934/jimo.2016.12.1507 |
[18] |
Wanbin Tong, Hongjin He, Chen Ling, Liqun Qi. A nonmonotone spectral projected gradient method for tensor eigenvalue complementarity problems. Numerical Algebra, Control and Optimization, 2020, 10 (4) : 425-437. doi: 10.3934/naco.2020042 |
[19] |
Yanfei Wang, Qinghua Ma. A gradient method for regularizing retrieval of aerosol particle size distribution function. Journal of Industrial and Management Optimization, 2009, 5 (1) : 115-126. doi: 10.3934/jimo.2009.5.115 |
[20] |
Nam-Yong Lee, Bradley J. Lucier. Preconditioned conjugate gradient method for boundary artifact-free image deblurring. Inverse Problems and Imaging, 2016, 10 (1) : 195-225. doi: 10.3934/ipi.2016.10.195 |
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]