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A modified Nelder-Mead barrier method for constrained optimization
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. Google Scholar |
[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. Google Scholar |
[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 |
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