American Institute of Mathematical Sciences

Advanced
December  2021, 11(4): 665-676. doi: 10.3934/naco.2021003

An alternate gradient method for optimization problems with orthogonality constraints

Yanmei Sun 1, and  Yakui Huang 2,,

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  December 2021 Early access  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.

Keywords: Orthogonality constraint, Stiefel manifold, Alternate gradient method, Gradient projection, Gradient reflection.
Mathematics Subject Classification: Primary: 65F15, 65K05; Secondary: 90C06.
Citation: Yanmei Sun, Yakui Huang. An alternate gradient method for optimization problems with orthogonality constraints. Numerical Algebra, Control and Optimization, 2021, 11 (4) : 665-676. doi: 10.3934/naco.2021003
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.

Figure 1.  Numerical results of the AG method with different $ m $
Figure Options

Download full-size image

Download as PowerPoint slide

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
Table Options

Download as excel

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
Table Options

Download as excel

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
Table Options

Download as excel

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
Table Options

Download as excel

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]

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

  • PDF downloads (212)
  • HTML views (367)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy