doi: 10.3934/naco.2021003

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

* Corresponding author: Yakui Huang

Received  November 2020 Revised  December 2020 Published  January 2021

Fund Project: The second author is supported by NSFC grant 11701137

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.

Citation: Yanmei Sun, Yakui Huang. An alternate gradient method for optimization problems with orthogonality constraints. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2021003
References:
[1] P. A. AbsilR. Mahony and R. Sepulchre, Optimization Algorithms on Matrix Manifolds, Princeton University Press, Princeton, 2008.  doi: 10.1515/9781400830244.  Google Scholar
[2]

T. AbrudanJ. 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.  Google Scholar

[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.  Google Scholar

[4]

A. D'AspremontL. 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.  Google Scholar

[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.  Google Scholar

[6]

L. Eldén and H. Park, A procrustes problem on the Stiefel manifold, Numerische Mathematik, 82 (1999), 599-619.  doi: 10.1007/s002110050432.  Google Scholar

[7]

A. EdelmanT. 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.  Google Scholar

[8]

B. GaoX. LiuX. 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.  Google Scholar

[9]

J. HuB. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

X. LiuX. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

C. YangJ. 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.  Google Scholar

[18]

C. YangJ. 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.  Google Scholar

[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.  Google Scholar

[20]

H. ZouT. Hastie and R. Tibshirani, Sparse principal component analysis, Journal of Computational and Graphical Statistics, 15 (2006), 265-286.  doi: 10.1198/106186006X113430.  Google Scholar

[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.  Google Scholar

show all references

References:
[1] P. A. AbsilR. Mahony and R. Sepulchre, Optimization Algorithms on Matrix Manifolds, Princeton University Press, Princeton, 2008.  doi: 10.1515/9781400830244.  Google Scholar
[2]

T. AbrudanJ. 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.  Google Scholar

[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.  Google Scholar

[4]

A. D'AspremontL. 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.  Google Scholar

[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.  Google Scholar

[6]

L. Eldén and H. Park, A procrustes problem on the Stiefel manifold, Numerische Mathematik, 82 (1999), 599-619.  doi: 10.1007/s002110050432.  Google Scholar

[7]

A. EdelmanT. 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.  Google Scholar

[8]

B. GaoX. LiuX. 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.  Google Scholar

[9]

J. HuB. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

X. LiuX. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

C. YangJ. 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.  Google Scholar

[18]

C. YangJ. 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.  Google Scholar

[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.  Google Scholar

[20]

H. ZouT. Hastie and R. Tibshirani, Sparse principal component analysis, Journal of Computational and Graphical Statistics, 15 (2006), 265-286.  doi: 10.1198/106186006X113430.  Google Scholar

[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.  Google Scholar

Figure 1.  Numerical results of the AG method with different $ m $
Table 1.  The results of GP, GR and AG on problem (2) with $ n = 500 $, $ p = 6 $
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$ (m=2) $ 2.01 433.4 -91.646891 3.8447E-04 4.5582E-15
AG$ (m=3) $ 2.21 483.4 -91.642235 3.8370E-04 6.2797E-15
AG$ (m=5) $ 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$ (m=2) $ 2.01 433.4 -91.646891 3.8447E-04 4.5582E-15
AG$ (m=3) $ 2.21 483.4 -91.642235 3.8370E-04 6.2797E-15
AG$ (m=5) $ 2.34 503.0 -91.646891 3.8142E-04 6.1260E-15
Table 2.  Numerical results on problem (2) with $ n = 1000 $ and $ \delta = 1 $
cpu iter fun KKT violation feasibility
$ p=6 $
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
$ p=10 $
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
$ p=20 $
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
$ p=30 $
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
$ p=40 $
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
$ p=50 $
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
$ p=60 $
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
$ p=6 $
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
$ p=10 $
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
$ p=20 $
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
$ p=30 $
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
$ p=40 $
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
$ p=50 $
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
$ p=60 $
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
Table 3.  Numerical results on problem (2) with $ n = 1000 $ and $ \delta = 10 $
cpu iter fun KKT violation feasibility
$ p=6 $
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
$ p=10 $
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
$ p=20 $
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
$ p=30 $
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
$ p=40 $
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
$ p=50 $
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
$ p=6 $
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
$ p=10 $
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
$ p=20 $
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
$ p=30 $
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
$ p=40 $
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
$ p=50 $
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
Table 4.  Numerical results on Kohn-Sham total energy minimization problem
cpu iter fun KKT violation feasibility
co2, $ n=2103,p=8 $
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, $ n=2103,p=7 $
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, $ n=8047,p=15 $
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, $ n=2103,p=4 $
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, $ n=5709,p=37 $
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, $ n=2103,p=6 $
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, $ n=251,p=7 $
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, $ n=2103,p=8 $
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, $ n=2103,p=7 $
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, $ n=8047,p=15 $
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, $ n=2103,p=4 $
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, $ n=5709,p=37 $
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, $ n=2103,p=6 $
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, $ n=251,p=7 $
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]

Min Li. A three term Polak-Ribière-Polyak conjugate gradient method close to the memoryless BFGS quasi-Newton method. Journal of Industrial & Management Optimization, 2020, 16 (1) : 245-260. doi: 10.3934/jimo.2018149

[2]

Tuan Hiep Pham, Jérôme Laverne, Jean-Jacques Marigo. Stress gradient effects on the nucleation and propagation of cohesive cracks. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 557-584. doi: 10.3934/dcdss.2016012

[3]

Matthias Erbar, Jan Maas. Gradient flow structures for discrete porous medium equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1355-1374. doi: 10.3934/dcds.2014.34.1355

[4]

Andrea Cianchi, Adele Ferone. Improving sharp Sobolev type inequalities by optimal remainder gradient norms. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1363-1386. doi: 10.3934/cpaa.2012.11.1363

[5]

Hong Seng Sim, Wah June Leong, Chuei Yee Chen, Siti Nur Iqmal Ibrahim. Multi-step spectral gradient methods with modified weak secant relation for large scale unconstrained optimization. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 377-387. doi: 10.3934/naco.2018024

[6]

Deren Han, Zehui Jia, Yongzhong Song, David Z. W. Wang. An efficient projection method for nonlinear inverse problems with sparsity constraints. Inverse Problems & Imaging, 2016, 10 (3) : 689-709. doi: 10.3934/ipi.2016017

[7]

Boris Kramer, John R. Singler. A POD projection method for large-scale algebraic Riccati equations. Numerical Algebra, Control & Optimization, 2016, 6 (4) : 413-435. doi: 10.3934/naco.2016018

[8]

Ziteng Wang, Shu-Cherng Fang, Wenxun Xing. On constraint qualifications: Motivation, design and inter-relations. Journal of Industrial & Management Optimization, 2013, 9 (4) : 983-1001. doi: 10.3934/jimo.2013.9.983

[9]

M. Phani Sudheer, Ravi S. Nanjundiah, A. S. Vasudeva Murthy. Revisiting the slow manifold of the Lorenz-Krishnamurthy quintet. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1403-1416. doi: 10.3934/dcdsb.2006.6.1403

[10]

Marian Gidea, Rafael de la Llave, Tere M. Seara. A general mechanism of instability in Hamiltonian systems: Skipping along a normally hyperbolic invariant manifold. Discrete & Continuous Dynamical Systems - A, 2020, 40 (12) : 6795-6813. doi: 10.3934/dcds.2020166

[11]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271

[12]

Teddy Pichard. A moment closure based on a projection on the boundary of the realizability domain: 1D case. Kinetic & Related Models, 2020, 13 (6) : 1243-1280. doi: 10.3934/krm.2020045

[13]

Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327

[14]

Tao Wu, Yu Lei, Jiao Shi, Maoguo Gong. An evolutionary multiobjective method for low-rank and sparse matrix decomposition. Big Data & Information Analytics, 2017, 2 (1) : 23-37. doi: 10.3934/bdia.2017006

[15]

Petra Csomós, Hermann Mena. Fourier-splitting method for solving hyperbolic LQR problems. Numerical Algebra, Control & Optimization, 2018, 8 (1) : 17-46. doi: 10.3934/naco.2018002

[16]

Christina Surulescu, Nicolae Surulescu. Modeling and simulation of some cell dispersion problems by a nonparametric method. Mathematical Biosciences & Engineering, 2011, 8 (2) : 263-277. doi: 10.3934/mbe.2011.8.263

[17]

Manfred Einsiedler, Elon Lindenstrauss. On measures invariant under diagonalizable actions: the Rank-One case and the general Low-Entropy method. Journal of Modern Dynamics, 2008, 2 (1) : 83-128. doi: 10.3934/jmd.2008.2.83

[18]

Xiaomao Deng, Xiao-Chuan Cai, Jun Zou. A parallel space-time domain decomposition method for unsteady source inversion problems. Inverse Problems & Imaging, 2015, 9 (4) : 1069-1091. doi: 10.3934/ipi.2015.9.1069

[19]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056

 Impact Factor: 

Article outline

Figures and Tables

[Back to Top]