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Simulated annealing and genetic algorithm based method for a bi-level seru loading problem with worker assignment in seru production systems
An alternating linearization bundle method for a class of nonconvex optimization problem with inexact information
1. | School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China |
2. | School of Information Engineering, Dalian Ocean University, Dalian 116024, China |
3. | School of Finance, Zhejiang University of Finance and Economics, Hangzhou 310018, China |
We propose an alternating linearization bundle method for minimizing the sum of a nonconvex function and a convex function. The convex function is assumed to be "simple" in the sense that finding its proximal-like point is relatively easy. The nonconvex function is known through oracles which provide inexact information. The errors in function values and subgradient evaluations might be unknown, but are bounded by universal constants. We examine an alternating linearization bundle method in this setting and obtain reasonable convergence properties. Numerical results show the good performance of the method.
References:
[1] |
H. Attouch, J. Bolte, P. Redont and A. Soubeyran,
Proximal alternating minimization and projection methods for nonconvex problems: an approach based on the Kurdyka-Łojasiewicz inequality, Math. Oper. Res., 35 (2010), 438-457.
doi: 10.1287/moor.1100.0449. |
[2] |
D. P. Bertsekas, Nonlinear Programming, 2nd edition, Athena Scientific Optimization and Computation Series, Athena Scientific, Belmont, 1999. |
[3] |
J. Bolte, S. Sabach and M. Teboulle,
Proximal alternating linearized minimization for nonconvex and nonsmooth problems, Math. Program., 146 (2014), 459-494.
doi: 10.1007/s10107-013-0701-9. |
[4] |
R. Chartrand,
Exact reconstruction of sparse signals via nonconvex minimization, IEEE Signal Process. Lett., 14 (2007), 707-710.
doi: 10.1109/lsp.2007.898300. |
[5] |
A. Danilidis and P. Georgiev,
Approximate convexity and submonotonicity, J. Math. Anal. Appl., 291 (2004), 292-301.
doi: 10.1016/j.jmaa.2003.11.004. |
[6] |
G. Emiel and C. Sagastizábal,
Incremental-like bundle methods with application to energy planning, Comput. Optim. Appl., 46 (2010), 305-332.
doi: 10.1007/s10589-009-9288-8. |
[7] |
C. Ferrier, Bornes Duales de Problémes d' Optimisation Polynomiaux, PhD thesis, Laboratoire Approximation et Optimisation, Université Paul Sabatier, Toulouse, France, 1997. Google Scholar |
[8] |
D. Goldfarb, S. Ma and K. Scheinberg,
Fast alternating linearization methods for minimizing the sum of two convex funcions, Math. Program., 141 (2013), 349-382.
doi: 10.1007/s10107-012-0530-2. |
[9] |
W. Hare and C. Sagastizábal,
Computing proximal points of nonconvex functions, Math. Program., 116 (2009), 221-258.
doi: 10.1007/s10107-007-0124-6. |
[10] |
W. Hare and C. Sagastizábal,
A redistributed proximal bundle method for nonconvex optimization, SIAM J. Optim., 20 (2010), 2442-2473.
doi: 10.1137/090754595. |
[11] |
W. Hare, C. Sagastizábal and M. Solodov,
A proximal bundle method for nonsmooth nonconvex functions with inexact information, Comput. Optim. Appl., 63 (2016), 1-28.
doi: 10.1007/s10589-015-9762-4. |
[12] |
K. C. Kiwiel,
An algorithm for nonsmooth convex minimization with errors, Math. Comp., 45 (1985), 173-180.
doi: 10.2307/2008055. |
[13] |
K. C. Kiwiel,
An alternating linearization bundle method for convex optimization and nonlinear multicommodity flow problems, Math. Program., 130 (2011), 59-84.
doi: 10.1007/s10107-009-0327-0. |
[14] |
K. C. Kiwiel,
A proximal bundle method with approximate subgradient linearizations, SIAM J. Optim., 16 (2006), 1007-1023.
doi: 10.1137/040603929. |
[15] |
K. C. Kiwiel,
A proximal-projection bundle method for Lagrangian relaxation, including semidefinite programming, SIAM J. Optim., 17 (2006), 1015-1034.
doi: 10.1137/050639284. |
[16] |
K. C. Kiwiel, C. H. Rosa and A. Ruszczynski,
Proximal decomposition via alternating linearization, SIAM J. Optim., 9 (1999), 668-689.
doi: 10.1137/S1052623495288064. |
[17] |
D. Li, L. P. Pang and S. Chen,
A proximal alternating linearization method for nonconvex optimization problems, Optim. Methods Softw., 29 (2014), 771-785.
doi: 10.1080/10556788.2013.854358. |
[18] |
D. Li, L. P. Pang and Z. Q. Xia,
An approximate alternating linearization decomposition method, J. Appl. Math. & Informatics, 28 (2010), 1249-1262.
|
[19] |
L. Lukšan and J. Vlček, Test problems for nonsmooth unconstrained and linearly constrained optimization, Technical Report No. 798, Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague, 2000. Google Scholar |
[20] |
J. Lv, L. P. Pang and F. Y. Meng,
A proximal bundle method for constrained nonsmooth nonconvex optimization with inexact information, J. Global Optim., 70 (2018), 517-549.
doi: 10.1007/s10898-017-0565-2. |
[21] |
J. Lv, L. P. Pang, N. Xu and Z. H. Xiao,
An infeasible bundle method for nonconvex constrained optimization with application to semi-infinite programming problems, Numer. Algorithms, 80 (2019), 397-427.
doi: 10.1007/s11075-018-0490-6. |
[22] |
F. Y. Meng, L. P. Pang, J. Lv and J. H. Wang,
An approximate bundle method for solving nonsmooth equilibrium problems, J. Global Optim., 68 (2017), 537-562.
doi: 10.1007/s10898-016-0490-9. |
[23] |
D. Noll, Bundle method for non-convex minimization with inexact subgradients and function values, in Computational and Analytical Mathematics, Springer Proceedings in Mathematics & Statistics, 50, Springer, New York, 2013, 555–592.
doi: 10.1007/978-1-4614-7621-4_26. |
[24] |
W. Oliveira, C. Sagastiz |
[25] |
L. P. Pang, J. Lv and J. H. Wang,
Constrained incremental bundle method with partial inexact oracle for nonsmooth convex semi-infinite programming problems, Comput. Optim. Appl., 64 (2016), 433-465.
doi: 10.1007/s10589-015-9810-0. |
[26] |
R. T. Rockafellar and R. J. B. Wets, Variational Analysis, Springer, Berlin, 1998.
doi: 10.1007/978-3-642-02431-3. |
[27] |
M. V. Solodov,
On approximations with finite precision in bundle methods for nonsmooth optimization, J. Optim. Theory Appl., 119 (2003), 151-165.
doi: 10.1023/B:JOTA.0000005046.70410.02. |
[28] |
J. E. Springarn,
Submonotone subdifferentials of Lipschitz functions, Trans. Amer. Math. Soc, 264 (1981), 77-89.
doi: 10.2307/1998411. |
[29] |
C. M. Tang, J. B. Jian and G. Y. Li,
A proximal-projection partial bundle method for convex constrained minimax problems, J. Ind. Manag. Optim., 15 (2019), 757-774.
doi: 10.3934/jimo.2018069. |
[30] |
C. M. Tang, J. M. Lv and J. B. Jian,
An alternating linearization bundle method for a class of nonconvex nonsmooth optimization problems, J. Inequal. Appl., 101 (2018), 1-23.
doi: 10.1186/s13660-018-1683-1. |
[31] |
Y. Yang, L. P. Pang, X. F. Ma and J. Shen,
Constrained nonconvex nonsmooth optimization via proximal bundle method, J. Optim. Theory Appl., 163 (2014), 900-925.
doi: 10.1007/s10957-014-0523-9. |
show all references
References:
[1] |
H. Attouch, J. Bolte, P. Redont and A. Soubeyran,
Proximal alternating minimization and projection methods for nonconvex problems: an approach based on the Kurdyka-Łojasiewicz inequality, Math. Oper. Res., 35 (2010), 438-457.
doi: 10.1287/moor.1100.0449. |
[2] |
D. P. Bertsekas, Nonlinear Programming, 2nd edition, Athena Scientific Optimization and Computation Series, Athena Scientific, Belmont, 1999. |
[3] |
J. Bolte, S. Sabach and M. Teboulle,
Proximal alternating linearized minimization for nonconvex and nonsmooth problems, Math. Program., 146 (2014), 459-494.
doi: 10.1007/s10107-013-0701-9. |
[4] |
R. Chartrand,
Exact reconstruction of sparse signals via nonconvex minimization, IEEE Signal Process. Lett., 14 (2007), 707-710.
doi: 10.1109/lsp.2007.898300. |
[5] |
A. Danilidis and P. Georgiev,
Approximate convexity and submonotonicity, J. Math. Anal. Appl., 291 (2004), 292-301.
doi: 10.1016/j.jmaa.2003.11.004. |
[6] |
G. Emiel and C. Sagastizábal,
Incremental-like bundle methods with application to energy planning, Comput. Optim. Appl., 46 (2010), 305-332.
doi: 10.1007/s10589-009-9288-8. |
[7] |
C. Ferrier, Bornes Duales de Problémes d' Optimisation Polynomiaux, PhD thesis, Laboratoire Approximation et Optimisation, Université Paul Sabatier, Toulouse, France, 1997. Google Scholar |
[8] |
D. Goldfarb, S. Ma and K. Scheinberg,
Fast alternating linearization methods for minimizing the sum of two convex funcions, Math. Program., 141 (2013), 349-382.
doi: 10.1007/s10107-012-0530-2. |
[9] |
W. Hare and C. Sagastizábal,
Computing proximal points of nonconvex functions, Math. Program., 116 (2009), 221-258.
doi: 10.1007/s10107-007-0124-6. |
[10] |
W. Hare and C. Sagastizábal,
A redistributed proximal bundle method for nonconvex optimization, SIAM J. Optim., 20 (2010), 2442-2473.
doi: 10.1137/090754595. |
[11] |
W. Hare, C. Sagastizábal and M. Solodov,
A proximal bundle method for nonsmooth nonconvex functions with inexact information, Comput. Optim. Appl., 63 (2016), 1-28.
doi: 10.1007/s10589-015-9762-4. |
[12] |
K. C. Kiwiel,
An algorithm for nonsmooth convex minimization with errors, Math. Comp., 45 (1985), 173-180.
doi: 10.2307/2008055. |
[13] |
K. C. Kiwiel,
An alternating linearization bundle method for convex optimization and nonlinear multicommodity flow problems, Math. Program., 130 (2011), 59-84.
doi: 10.1007/s10107-009-0327-0. |
[14] |
K. C. Kiwiel,
A proximal bundle method with approximate subgradient linearizations, SIAM J. Optim., 16 (2006), 1007-1023.
doi: 10.1137/040603929. |
[15] |
K. C. Kiwiel,
A proximal-projection bundle method for Lagrangian relaxation, including semidefinite programming, SIAM J. Optim., 17 (2006), 1015-1034.
doi: 10.1137/050639284. |
[16] |
K. C. Kiwiel, C. H. Rosa and A. Ruszczynski,
Proximal decomposition via alternating linearization, SIAM J. Optim., 9 (1999), 668-689.
doi: 10.1137/S1052623495288064. |
[17] |
D. Li, L. P. Pang and S. Chen,
A proximal alternating linearization method for nonconvex optimization problems, Optim. Methods Softw., 29 (2014), 771-785.
doi: 10.1080/10556788.2013.854358. |
[18] |
D. Li, L. P. Pang and Z. Q. Xia,
An approximate alternating linearization decomposition method, J. Appl. Math. & Informatics, 28 (2010), 1249-1262.
|
[19] |
L. Lukšan and J. Vlček, Test problems for nonsmooth unconstrained and linearly constrained optimization, Technical Report No. 798, Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague, 2000. Google Scholar |
[20] |
J. Lv, L. P. Pang and F. Y. Meng,
A proximal bundle method for constrained nonsmooth nonconvex optimization with inexact information, J. Global Optim., 70 (2018), 517-549.
doi: 10.1007/s10898-017-0565-2. |
[21] |
J. Lv, L. P. Pang, N. Xu and Z. H. Xiao,
An infeasible bundle method for nonconvex constrained optimization with application to semi-infinite programming problems, Numer. Algorithms, 80 (2019), 397-427.
doi: 10.1007/s11075-018-0490-6. |
[22] |
F. Y. Meng, L. P. Pang, J. Lv and J. H. Wang,
An approximate bundle method for solving nonsmooth equilibrium problems, J. Global Optim., 68 (2017), 537-562.
doi: 10.1007/s10898-016-0490-9. |
[23] |
D. Noll, Bundle method for non-convex minimization with inexact subgradients and function values, in Computational and Analytical Mathematics, Springer Proceedings in Mathematics & Statistics, 50, Springer, New York, 2013, 555–592.
doi: 10.1007/978-1-4614-7621-4_26. |
[24] |
W. Oliveira, C. Sagastiz |
[25] |
L. P. Pang, J. Lv and J. H. Wang,
Constrained incremental bundle method with partial inexact oracle for nonsmooth convex semi-infinite programming problems, Comput. Optim. Appl., 64 (2016), 433-465.
doi: 10.1007/s10589-015-9810-0. |
[26] |
R. T. Rockafellar and R. J. B. Wets, Variational Analysis, Springer, Berlin, 1998.
doi: 10.1007/978-3-642-02431-3. |
[27] |
M. V. Solodov,
On approximations with finite precision in bundle methods for nonsmooth optimization, J. Optim. Theory Appl., 119 (2003), 151-165.
doi: 10.1023/B:JOTA.0000005046.70410.02. |
[28] |
J. E. Springarn,
Submonotone subdifferentials of Lipschitz functions, Trans. Amer. Math. Soc, 264 (1981), 77-89.
doi: 10.2307/1998411. |
[29] |
C. M. Tang, J. B. Jian and G. Y. Li,
A proximal-projection partial bundle method for convex constrained minimax problems, J. Ind. Manag. Optim., 15 (2019), 757-774.
doi: 10.3934/jimo.2018069. |
[30] |
C. M. Tang, J. M. Lv and J. B. Jian,
An alternating linearization bundle method for a class of nonconvex nonsmooth optimization problems, J. Inequal. Appl., 101 (2018), 1-23.
doi: 10.1186/s13660-018-1683-1. |
[31] |
Y. Yang, L. P. Pang, X. F. Ma and J. Shen,
Constrained nonconvex nonsmooth optimization via proximal bundle method, J. Optim. Theory Appl., 163 (2014), 900-925.
doi: 10.1007/s10957-014-0523-9. |



Algorithm 1 An alternating linearization bundle algorithm |
step 0 (Initialization) Select a starting point |
step 1 (Solving the |
step 2 (Solving the |
Compute |
step 3 (Stopping criterion) Compute |
step 4 (Descent test) Compute |
step 5 (Bundle update and loop) For a proximal parameter |
Algorithm 1 An alternating linearization bundle algorithm |
step 0 (Initialization) Select a starting point |
step 1 (Solving the |
step 2 (Solving the |
Compute |
step 3 (Stopping criterion) Compute |
step 4 (Descent test) Compute |
step 5 (Bundle update and loop) For a proximal parameter |
Alg. 1 | 2 | 3.342427 | 2 | 1 | 2 | |
3.343146 | 2 | 1 | 2 | |||
3.343146 | 2 | 1 | 2 | |||
Alg. 1 | 2 | 24.477350 | 7 | 4 | 7 | |
24.479795 | 9 | 4 | 9 | |||
24.479795 | 11 | 8 | 11 | |||
Alg. 1 | 2 | -0.998473 | 13 | 10 | 13 | |
-0.999989 | 15 | 12 | 15 | |||
-0.999989 | 18 | 17 | 18 | |||
Alg. 1 | 2 | 48.153612 | 4 | 3 | 4 | |
48.153612 | 3 | 2 | 3 | |||
48.153612 | 4 | 2 | 4 | |||
Alg. 1 | 4 | 39.698418 | 6 | 5 | 6 | |
39.715617 | 7 | 6 | 7 | |||
39.715617 | 9 | 8 | 9 | |||
Alg. 1 | 5 | 50.250278 | 4 | 3 | 4 | |
50.250278 | 5 | 1 | 5 | |||
50.250278 | 4 | 1 | 4 | |||
Alg. 1 | 20 | 0.557216 | 17 | 7 | 17 | |
0.552786 | 19 | 6 | 19 | |||
0.552786 | 22 | 2 | 22 |
Alg. 1 | 2 | 3.342427 | 2 | 1 | 2 | |
3.343146 | 2 | 1 | 2 | |||
3.343146 | 2 | 1 | 2 | |||
Alg. 1 | 2 | 24.477350 | 7 | 4 | 7 | |
24.479795 | 9 | 4 | 9 | |||
24.479795 | 11 | 8 | 11 | |||
Alg. 1 | 2 | -0.998473 | 13 | 10 | 13 | |
-0.999989 | 15 | 12 | 15 | |||
-0.999989 | 18 | 17 | 18 | |||
Alg. 1 | 2 | 48.153612 | 4 | 3 | 4 | |
48.153612 | 3 | 2 | 3 | |||
48.153612 | 4 | 2 | 4 | |||
Alg. 1 | 4 | 39.698418 | 6 | 5 | 6 | |
39.715617 | 7 | 6 | 7 | |||
39.715617 | 9 | 8 | 9 | |||
Alg. 1 | 5 | 50.250278 | 4 | 3 | 4 | |
50.250278 | 5 | 1 | 5 | |||
50.250278 | 4 | 1 | 4 | |||
Alg. 1 | 20 | 0.557216 | 17 | 7 | 17 | |
0.552786 | 19 | 6 | 19 | |||
0.552786 | 22 | 2 | 22 |
5 | Alg. 1 | 4.22e-007 | 28 | 28 | 0.5833 |
3.13e-007 | 30 | 30 | 0.6032 | ||
2.37e-007 | 31 | 31 | 0.6817 | ||
6 | Alg. 1 | 1.88e-007 | 27 | 28 | 0.5024 |
2.39e-007 | 27 | 30 | 0.5924 | ||
1.43e-006 | 27 | 33 | 1.0164 | ||
7 | Alg. 1 | 3.82e-005 | 33 | 36 | 0.5437 |
4.83e-007 | 32 | 37 | 0.5771 | ||
3.34e-006 | 32 | 34 | 1.4873 | ||
8 | Alg. 1 | 4.16e-005 | 38 | 41 | 0.8272 |
4.63e-005 | 37 | 43 | 0.9136 | ||
4.25e-003 | 43 | 48 | 5.7492 | ||
9 | Alg. 1 | 5.18e-005 | 33 | 36 | 1.1306 |
6.48e-006 | 37 | 39 | 1.3848 | ||
2.41e-004 | 37 | 44 | 2.1447 | ||
10 | Alg. 1 | 4.96e-005 | 38 | 43 | 0.9137 |
4.17e-006 | 37 | 46 | 0.9747 | ||
5.43e-005 | 44 | 57 | 4.0254 | ||
11 | Alg. 1 | 4.37e-005 | 48 | 55 | 1.0873 |
2.73e-006 | 47 | 56 | 1.1473 | ||
4.67e-005 | 62 | 69 | 1.2734 | ||
12 | Alg. 1 | 4.37e-006 | 48 | 65 | 1.2063 |
5.46e-006 | 51 | 67 | 1.3593 | ||
4.73e-006 | 74 | 80 | 1.4932 | ||
13 | Alg. 1 | 4.58e-005 | 48 | 68 | 1.1283 |
3.28e-006 | 54 | 72 | 1.2863 | ||
8.53e-005 | 82 | 93 | 2.3602 | ||
14 | Alg. 1 | 4.63e-005 | 56 | 65 | 1.4853 |
3.17e-005 | 54 | 68 | 1.7437 | ||
3.14e-006 | 63 | 72 | 2.4185 | ||
15 | Alg. 1 | 1.16e-004 | 46 | 58 | 1.0673 |
2.41e-004 | 52 | 64 | 1.1536 | ||
3.38e-003 | 57 | 72 | 1.2183 | ||
20 | Alg. 1 | 4.62e-004 | 81 | 96 | 4.4328 |
3.84e-003 | 79 | 95 | 4.8753 | ||
0.0198 | 104 | 131 | 10.7273 |
5 | Alg. 1 | 4.22e-007 | 28 | 28 | 0.5833 |
3.13e-007 | 30 | 30 | 0.6032 | ||
2.37e-007 | 31 | 31 | 0.6817 | ||
6 | Alg. 1 | 1.88e-007 | 27 | 28 | 0.5024 |
2.39e-007 | 27 | 30 | 0.5924 | ||
1.43e-006 | 27 | 33 | 1.0164 | ||
7 | Alg. 1 | 3.82e-005 | 33 | 36 | 0.5437 |
4.83e-007 | 32 | 37 | 0.5771 | ||
3.34e-006 | 32 | 34 | 1.4873 | ||
8 | Alg. 1 | 4.16e-005 | 38 | 41 | 0.8272 |
4.63e-005 | 37 | 43 | 0.9136 | ||
4.25e-003 | 43 | 48 | 5.7492 | ||
9 | Alg. 1 | 5.18e-005 | 33 | 36 | 1.1306 |
6.48e-006 | 37 | 39 | 1.3848 | ||
2.41e-004 | 37 | 44 | 2.1447 | ||
10 | Alg. 1 | 4.96e-005 | 38 | 43 | 0.9137 |
4.17e-006 | 37 | 46 | 0.9747 | ||
5.43e-005 | 44 | 57 | 4.0254 | ||
11 | Alg. 1 | 4.37e-005 | 48 | 55 | 1.0873 |
2.73e-006 | 47 | 56 | 1.1473 | ||
4.67e-005 | 62 | 69 | 1.2734 | ||
12 | Alg. 1 | 4.37e-006 | 48 | 65 | 1.2063 |
5.46e-006 | 51 | 67 | 1.3593 | ||
4.73e-006 | 74 | 80 | 1.4932 | ||
13 | Alg. 1 | 4.58e-005 | 48 | 68 | 1.1283 |
3.28e-006 | 54 | 72 | 1.2863 | ||
8.53e-005 | 82 | 93 | 2.3602 | ||
14 | Alg. 1 | 4.63e-005 | 56 | 65 | 1.4853 |
3.17e-005 | 54 | 68 | 1.7437 | ||
3.14e-006 | 63 | 72 | 2.4185 | ||
15 | Alg. 1 | 1.16e-004 | 46 | 58 | 1.0673 |
2.41e-004 | 52 | 64 | 1.1536 | ||
3.38e-003 | 57 | 72 | 1.2183 | ||
20 | Alg. 1 | 4.62e-004 | 81 | 96 | 4.4328 |
3.84e-003 | 79 | 95 | 4.8753 | ||
0.0198 | 104 | 131 | 10.7273 |
5 | Alg. 1 | 2.18e-004 | 21 | 26 | 0.2063 |
3.86e-004 | 19 | 26 | 0.2165 | ||
5.74e-004 | 23 | 31 | 0.6639 | ||
6 | Alg. 1 | 3.65e-003 | 27 | 33 | 0.3452 |
3.61e-003 | 24 | 29 | 0.3277 | ||
3.46e-003 | 22 | 31 | 0.3532 | ||
7 | Alg. 1 | 2.84e-003 | 36 | 41 | 1.5834 |
2.65e-004 | 43 | 56 | 1.9734 | ||
2.14e-003 | 56 | 74 | 5.8017 | ||
8 | Alg. 1 | 2.88e-004 | 24 | 33 | 0.4110 |
|
3.28e-004 | 29 | 36 | 0.4368 | |
1.29e-003 | 31 | 40 | 1.2455 | ||
9 | Alg. 1 | 2.46e-004 | 29 | 53 | 1.1681 |
5.49e-004 | 32 | 54 | 1.2518 | ||
2.74e-003 | 36 | 45 | 1.8386 | ||
10 | Alg. 1 | 3.48e-003 | 19 | 34 | 1.0538 |
2.86e-003 | 21 | 36 | 1.2023 | ||
1.55e-002 | 63 | 88 | 7.2973 | ||
11 | Alg. 1 | 5.75e-003 | 36 | 57 | 1.3976 |
8.64e-003 | 37 | 55 | 1.4683 | ||
1.17e-002 | 64 | 81 | 11.2496 | ||
12 | Alg. 1 | 1.32e-002 | 29 | 54 | 1.3056 |
1.22e-002 | 29 | 57 | 1.3205 | ||
1.24e-002 | 64 | 79 | 1.6228 | ||
13 | Alg. 1 | 4.16e-003 | 35 | 71 | 1.5946 |
3.144-004 | 38 | 63 | 1.6634 | ||
2.84e-003 | 67 | 83 | 2.1066 | ||
14 | Alg. 1 | 3.53e-003 | 30 | 46 | 1.2391 |
3.85e-003 | 32 | 48 | 1.3884 | ||
7.95e-003 | 58 | 81 | 2.4639 | ||
15 | Alg. 1 | 2.43e-002 | 41 | 67 | 0.9864 |
1.41e-003 | 49 | 76 | 1.0356 | ||
4.63e-002 | 66 | 84 | 3.7320 | ||
20 | Alg. 1 | 1.53e-004 | 33 | 51 | 1.4601 |
3.74e-004 | 35 | 56 | 1.6054 | ||
7.78e-002 | 87 | 119 | 5.8107 |
5 | Alg. 1 | 2.18e-004 | 21 | 26 | 0.2063 |
3.86e-004 | 19 | 26 | 0.2165 | ||
5.74e-004 | 23 | 31 | 0.6639 | ||
6 | Alg. 1 | 3.65e-003 | 27 | 33 | 0.3452 |
3.61e-003 | 24 | 29 | 0.3277 | ||
3.46e-003 | 22 | 31 | 0.3532 | ||
7 | Alg. 1 | 2.84e-003 | 36 | 41 | 1.5834 |
2.65e-004 | 43 | 56 | 1.9734 | ||
2.14e-003 | 56 | 74 | 5.8017 | ||
8 | Alg. 1 | 2.88e-004 | 24 | 33 | 0.4110 |
|
3.28e-004 | 29 | 36 | 0.4368 | |
1.29e-003 | 31 | 40 | 1.2455 | ||
9 | Alg. 1 | 2.46e-004 | 29 | 53 | 1.1681 |
5.49e-004 | 32 | 54 | 1.2518 | ||
2.74e-003 | 36 | 45 | 1.8386 | ||
10 | Alg. 1 | 3.48e-003 | 19 | 34 | 1.0538 |
2.86e-003 | 21 | 36 | 1.2023 | ||
1.55e-002 | 63 | 88 | 7.2973 | ||
11 | Alg. 1 | 5.75e-003 | 36 | 57 | 1.3976 |
8.64e-003 | 37 | 55 | 1.4683 | ||
1.17e-002 | 64 | 81 | 11.2496 | ||
12 | Alg. 1 | 1.32e-002 | 29 | 54 | 1.3056 |
1.22e-002 | 29 | 57 | 1.3205 | ||
1.24e-002 | 64 | 79 | 1.6228 | ||
13 | Alg. 1 | 4.16e-003 | 35 | 71 | 1.5946 |
3.144-004 | 38 | 63 | 1.6634 | ||
2.84e-003 | 67 | 83 | 2.1066 | ||
14 | Alg. 1 | 3.53e-003 | 30 | 46 | 1.2391 |
3.85e-003 | 32 | 48 | 1.3884 | ||
7.95e-003 | 58 | 81 | 2.4639 | ||
15 | Alg. 1 | 2.43e-002 | 41 | 67 | 0.9864 |
1.41e-003 | 49 | 76 | 1.0356 | ||
4.63e-002 | 66 | 84 | 3.7320 | ||
20 | Alg. 1 | 1.53e-004 | 33 | 51 | 1.4601 |
3.74e-004 | 35 | 56 | 1.6054 | ||
7.78e-002 | 87 | 119 | 5.8107 |
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