Extremely large computation time in a small non convex problem
回答済みI am trying to solve a principal components problem in Gurobi. This problem is defined as:
$$ max_w(w^tX^tXw) \quad s.t. \quad \|w\|_2=1 $$
I have programmed the following function that solves this problem in python:
# Load packages
import numpy as np
import sklearn.datasets
import gurobipy as gp
from gurobipy import GRB
from itertools import product
from time import time
# Define PCA in Gurobi
def pca_gurobi(x):
# Center matrix
x_center = x - np.mean(x, axis=0)
n, p = x.shape
# Compute variability
Quad = np.dot(x_center.T, x_center)
pca_gurobi = gp.Model()
weights = pca_gurobi.addVars(p, lb=-GRB.INFINITY, name="weights") # pca loading
obj_pca = (1/n) * sum(Quad[i, j] * weights[i] * weights[j] for i, j in product(range(p), repeat=2))
pca_gurobi.addConstr(sum(weights[i]**2 for i in range(p)) == 1)
# Solve as a non convex optimization problem
pca_gurobi.setObjective(obj_pca, GRB.MAXIMIZE)
pca_gurobi.params.NonConvex = 2 # Let gurobi know that constraint is non convex
pca_gurobi.params.OutputFlag = 0
pca_gurobi.optimize()
weights_pca_gurobi = np.array([weights[i].X for i in range(p)])
return weights_pca_gurobi
And I tested this function on a synthetic dataset with 10 variables, achieving the expected numerical solution in 1 second:
# Generate 10 variables dataset
x, y, true_beta = sklearn.datasets.make_regression(n_samples=500, n_features=10, n_informative=10, n_targets=1,
noise=1.0, shuffle=True, coef=True, random_state=1)
start_time = time()
pca_gurobi(x)
print(f'Execution time: {np.round(time() - start_time, 2)} seconds')
Execution time: 1.74 seconds
The problem comes here. A small increase in the number of variables in the dataset, from 10 variables to 20 variables, produces a problem that Gurobi is not able to solve. I left the following code executing for more than 12 hours before I decided to stop the execution, it does not yield any error message but nor provides a solution, it simply keeps executing.
# Generate 20 variables dataset
x, y, true_beta = sklearn.datasets.make_regression(n_samples=500, n_features=20, n_informative=10, n_targets=1,
noise=1.0, shuffle=True, coef=True, random_state=1)
start_time = time()
pca_gurobi(x)
print(f'Execution time: {np.round(time() - start_time, 2)} seconds')
I do not understand what is happening here. The problem is simple enough and the number of Gurobi variables is still small (20). Is there any way of reformulating this problem so that it can be solved? Am I doing something wrong?
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正式なコメント
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Gurobi Staff
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Hi Alvaro,
Could you post the logs of the two runs? If possible, could you shorten the B&B part to show only relevant parts to make the logs more compact and easier to read?
In general, it is often the case for nonconvex problems that even a seemingly slight change may decide between the problem being solved in a few seconds and not being solved within hours.Your data is generated randomly, correct? It is also possible that the data generated in the 10 variables setting strongly contributes to the good solution process, while the data generated in the 20 variables case results in bad numerical behavior.
Did you test what happens when you generate problems with 11-19 variables? Do you see a steady increase in runtime over the number of variables or does the runtime suddenly increase drastically?
Did you check whether the formulated model is correct? You can do this quite easily with the help of the Model.write() function to generate an LP file which you can analyze.
Best regards,
Jaromił0 -
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Hello and thank you for your answer,
Yes, the data is generated randomly, but I used different ways of generating different datasets and found the same problem. As I increase the number of variables, the computation time greatly increases.
Actually, as I increase the number of variables, I see an increase of the computation time of roughly 4 times the previus time per variable added. Here are the computation times for 10 variables up to 16:
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10 -> Execution time: 1.74 seconds
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11 -> Execution time: 3.56 seconds
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12 -> Execution time: 32 seconds
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13 -> Execution time: 207.01 seconds
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14 -> Execution time: 117.98 seconds
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15 -> Execution time: 640.82 seconds
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16 -> Execution time: 2357.12 seconds
I am 100% sure that the solutions achieved are correct so the model is correctly formulated. However the computation time is too large, I am not sure if there would be another way of formulationg the same problem achieving the results faster.
I am sorry, I do not know what do you refer with B&B so I will simply post the full logs, but If you can tell me what are the relevant parts you are interested in, I will gladly edit the answer to send just that.
Log using 10 variables
... # Generate 10 variables dataset
... x, y, true_beta = sklearn.datasets.make_regression(n_samples=500, n_features=10, n_informative=10, n_targets=1,
... noise=1.0, shuffle=True, coef=True, random_state=1)
...
... start_time = time()
... pca_gurobi(x)
... print(f'Execution time: {np.round(time() - start_time, 2)} seconds')
...
Academic license - for non-commercial use only - expires 2021-01-30
Using license file C:\Users\alvaromc317\gurobi.lic
Changed value of parameter NonConvex to 2
Prev: -1 Min: -1 Max: 2 Default: -1
Gurobi Optimizer version 9.1.1 build v9.1.1rc0 (win64)
Thread count: 6 physical cores, 12 logical processors, using up to 12 threads
Optimize a model with 0 rows, 10 columns and 0 nonzeros
Model fingerprint: 0x5751426a
Model has 55 quadratic objective terms
Model has 1 quadratic constraint
Coefficient statistics:
Matrix range [0e+00, 0e+00]
QMatrix range [1e+00, 1e+00]
Objective range [0e+00, 0e+00]
QObjective range [3e-03, 2e+00]
Bounds range [0e+00, 0e+00]
RHS range [0e+00, 0e+00]
QRHS range [1e+00, 1e+00]
Continuous model is non-convex -- solving as a MIP.
Presolve time: 0.01s
Presolved: 122 rows, 66 columns, 385 nonzeros
Presolved model has 55 bilinear constraint(s)
Variable types: 66 continuous, 0 integer (0 binary)
Root relaxation: objective 5.058420e+00, 12 iterations, 0.00 seconds
Nodes | Current Node | Objective Bounds | Work
Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time
0 0 5.05842 0 46 - 5.05842 - - 0s
0 0 1.44751 0 31 - 1.44751 - - 0s
0 0 1.38838 0 28 - 1.38838 - - 0s
0 0 1.38631 0 36 - 1.38631 - - 0s
0 0 1.37635 0 34 - 1.37635 - - 0s
0 0 1.36429 0 35 - 1.36429 - - 0s
0 0 1.36266 0 32 - 1.36266 - - 0s
0 0 1.35691 0 35 - 1.35691 - - 0s
H 0 0 1.2253735 1.35691 10.7% - 0s
0 2 1.35691 0 35 1.22537 1.35691 10.7% - 0s
* 521 502 47 1.2757249 1.33108 4.34% 12.9 0s
* 545 482 50 1.2759028 1.33108 4.32% 12.7 0s
* 546 482 50 1.2759029 1.33108 4.32% 12.7 0s
* 919 415 57 1.2789812 1.32726 3.78% 14.2 0s
* 1903 643 41 1.2790740 1.28082 0.14% 19.0 0s
* 2015 635 40 1.2790877 1.28082 0.14% 19.3 0s
* 2556 829 41 1.2792389 1.28040 0.09% 19.9 0s
* 2866 844 47 1.2792390 1.28040 0.09% 19.5 0s
* 2867 844 46 1.2792432 1.28040 0.09% 19.5 0s
* 3085 820 57 1.2797701 1.28040 0.05% 19.4 0s
* 3599 776 45 1.2797777 1.28040 0.05% 19.1 0s
* 3683 776 49 1.2797783 1.28040 0.05% 19.1 0s
* 3778 745 64 1.2798498 1.28040 0.04% 19.0 0s
* 6492 1173 72 1.2798612 1.28010 0.02% 18.5 0s
* 7256 1438 74 1.2798683 1.28008 0.02% 18.4 0s
*10977 2859 74 1.2798707 1.28004 0.01% 17.6 1s
Cutting planes:
PSD: 158
Explored 20602 nodes (338360 simplex iterations) in 1.81 seconds
Thread count was 12 (of 12 available processors)
Solution count 10: 1.27987 1.27987 1.27986 ... 1.27924
Optimal solution found (tolerance 1.00e-04)
Warning: max constraint violation (4.3771e-06) exceeds tolerance
Best objective 1.279873537698e+00, best bound 1.279993052390e+00, gap 0.0093%
Execution time: 1.84 secondsLog using 16 variables
x, y, true_beta = sklearn.datasets.make_regression(n_samples=500, n_features=16, n_informative=16, n_targets=1,
... noise=1.0, shuffle=True, coef=True, random_state=1)
...
... start_time = time()
... pca_gurobi(x)
... print(f'Execution time: {np.round(time() - start_time, 2)} seconds')
...
Changed value of parameter NonConvex to 2
Prev: -1 Min: -1 Max: 2 Default: -1
Gurobi Optimizer version 9.1.1 build v9.1.1rc0 (win64)
Thread count: 6 physical cores, 12 logical processors, using up to 12 threads
Optimize a model with 0 rows, 16 columns and 0 nonzeros
Model fingerprint: 0x04193c8c
Model has 136 quadratic objective terms
Model has 1 quadratic constraint
Coefficient statistics:
Matrix range [0e+00, 0e+00]
QMatrix range [1e+00, 1e+00]
Objective range [0e+00, 0e+00]
QObjective range [1e-03, 2e+00]
Bounds range [0e+00, 0e+00]
RHS range [0e+00, 0e+00]
QRHS range [1e+00, 1e+00]
Continuous model is non-convex -- solving as a MIP.
Presolve time: 0.00s
Presolved: 290 rows, 153 columns, 952 nonzeros
Presolved model has 136 bilinear constraint(s)
Variable types: 153 continuous, 0 integer (0 binary)
Root relaxation: objective 1.022745e+01, 63 iterations, 0.00 seconds
Nodes | Current Node | Objective Bounds | Work
Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time
0 0 10.22745 0 121 - 10.22745 - - 0s
0 0 1.65561 0 74 - 1.65561 - - 0s
0 0 1.54693 0 85 - 1.54693 - - 0s
0 0 1.52246 0 79 - 1.52246 - - 0s
0 0 1.52092 0 82 - 1.52092 - - 0s
0 0 1.50626 0 75 - 1.50626 - - 0s
H 0 0 1.2698936 1.50626 18.6% - 0s
0 2 1.50626 0 75 1.26989 1.50626 18.6% - 0s
* 906 725 103 1.3535117 1.47976 9.33% 32.3 0s
* 907 725 103 1.3535120 1.47976 9.33% 32.3 0s
* 2155 977 57 1.3536847 1.36352 0.73% 51.2 2s
* 2252 995 57 1.3539251 1.36352 0.71% 51.5 2s
* 2413 998 51 1.3539683 1.36352 0.71% 51.4 2s
* 2511 1017 78 1.3567543 1.36352 0.50% 50.9 2s
* 2514 982 79 1.3567547 1.36352 0.50% 50.8 2s
* 3372 1136 70 1.3568061 1.36352 0.49% 51.3 2s
* 3810 1241 89 1.3575212 1.36352 0.44% 49.4 2s
* 3813 1241 90 1.3575216 1.36352 0.44% 49.4 2s
* 4138 1323 89 1.3575573 1.36123 0.27% 48.0 2s
* 4240 1272 92 1.3575681 1.36110 0.26% 47.6 2s
* 4499 1120 84 1.3576843 1.36091 0.24% 47.3 2s
* 4501 1120 85 1.3576844 1.36091 0.24% 47.3 2s
* 5706 1408 99 1.3577363 1.35978 0.15% 48.2 3s
* 5712 1408 100 1.3577364 1.35978 0.15% 48.1 3s
9106 2541 1.35817 30 85 1.35774 1.35889 0.09% 54.2 5s
21064 6891 1.35799 37 74 1.35774 1.35846 0.05% 55.2 10s
32098 11540 cutoff 40 1.35774 1.35836 0.05% 54.1 15s
*36568 12732 94 1.3577419 1.35833 0.04% 53.9 16s
*36569 12713 95 1.3577421 1.35833 0.04% 53.9 16s
*40349 14038 95 1.3577491 1.35831 0.04% 53.2 18s
*40350 14026 95 1.3577496 1.35831 0.04% 53.2 18s
45189 16568 1.35776 39 73 1.35775 1.35829 0.04% 52.7 20s
*46063 16081 100 1.3577585 1.35829 0.04% 52.5 20s
*46071 16075 103 1.3577586 1.35829 0.04% 52.5 20s
*46072 16063 103 1.3577588 1.35829 0.04% 52.5 20s
57954 21305 cutoff 65 1.35776 1.35825 0.04% 51.6 25s
72866 27313 1.35807 44 80 1.35776 1.35823 0.03% 50.5 30s
87373 33373 cutoff 58 1.35776 1.35821 0.03% 49.7 36s
98318 38185 cutoff 46 1.35776 1.35819 0.03% 49.0 40s
111506 43426 1.35801 43 73 1.35776 1.35818 0.03% 48.2 45s
127569 48623 cutoff 55 1.35776 1.35817 0.03% 47.6 50s
142049 54376 1.35782 51 80 1.35776 1.35816 0.03% 47.0 55s
152664 58419 1.35793 37 61 1.35776 1.35815 0.03% 46.7 60s
165852 63516 1.35780 41 61 1.35776 1.35814 0.03% 46.3 65s
182222 70003 1.35789 56 86 1.35776 1.35814 0.03% 45.8 70s
196837 75354 1.35803 40 79 1.35776 1.35813 0.03% 45.4 75s
211191 81296 1.35779 57 83 1.35776 1.35812 0.03% 45.1 80s
223645 86020 cutoff 62 1.35776 1.35812 0.03% 44.8 85s
238111 91451 cutoff 35 1.35776 1.35811 0.03% 44.5 90s
252110 96977 1.35776 56 80 1.35776 1.35811 0.03% 44.2 95s
265244 102433 1.35795 46 64 1.35776 1.35811 0.03% 43.9 101s
275899 106736 1.35777 49 69 1.35776 1.35810 0.03% 43.7 105s
290504 112326 cutoff 40 1.35776 1.35810 0.03% 43.4 110s
306623 118943 1.35787 54 94 1.35776 1.35810 0.02% 43.2 115s
321282 125291 1.35780 53 90 1.35776 1.35809 0.02% 42.9 120s
335373 130002 1.35806 46 73 1.35776 1.35809 0.02% 42.6 125s
349590 134926 cutoff 58 1.35776 1.35809 0.02% 42.4 132s
354823 137051 1.35791 42 82 1.35776 1.35809 0.02% 42.3 136s
360594 139111 1.35799 49 86 1.35776 1.35808 0.02% 42.3 140s
375251 144185 1.35795 38 68 1.35776 1.35808 0.02% 42.1 145s
391656 149939 1.35781 55 74 1.35776 1.35808 0.02% 41.9 150s
406103 155506 1.35795 48 77 1.35776 1.35808 0.02% 41.7 155s
422374 161731 cutoff 47 1.35776 1.35807 0.02% 41.5 160s
436692 167164 cutoff 60 1.35776 1.35807 0.02% 41.3 165s
452509 172840 1.35782 44 83 1.35776 1.35807 0.02% 41.2 170s
468744 178591 1.35778 50 76 1.35776 1.35807 0.02% 41.0 175s
483259 183766 1.35784 56 81 1.35776 1.35806 0.02% 40.8 180s
493791 187590 1.35794 54 75 1.35776 1.35806 0.02% 40.8 185s
510266 193954 1.35783 42 70 1.35776 1.35806 0.02% 40.6 190s
526495 200454 1.35795 39 84 1.35776 1.35806 0.02% 40.4 195s
538966 204772 1.35789 49 81 1.35776 1.35806 0.02% 40.3 201s
551719 209739 cutoff 56 1.35776 1.35805 0.02% 40.3 205s
566204 214798 1.35778 67 67 1.35776 1.35805 0.02% 40.1 210s
582434 220760 1.35793 29 88 1.35776 1.35805 0.02% 40.0 215s
598670 227101 1.35788 60 86 1.35776 1.35805 0.02% 39.9 220s
614828 233161 cutoff 50 1.35776 1.35805 0.02% 39.8 225s
630997 239468 1.35792 45 65 1.35776 1.35805 0.02% 39.6 230s
642858 244053 1.35783 62 93 1.35776 1.35804 0.02% 39.6 235s
657854 249561 1.35783 60 74 1.35776 1.35804 0.02% 39.4 240s
674150 255947 1.35792 40 84 1.35776 1.35804 0.02% 39.3 245s
692017 261972 1.35800 51 80 1.35776 1.35804 0.02% 39.2 250s
708152 267500 cutoff 55 1.35776 1.35804 0.02% 39.1 255s
724091 273169 1.35777 72 74 1.35776 1.35804 0.02% 39.0 260s
736920 277990 cutoff 64 1.35776 1.35803 0.02% 38.9 265s
753190 283598 1.35793 36 86 1.35776 1.35803 0.02% 38.8 270s
769068 289582 1.35784 52 67 1.35776 1.35803 0.02% 38.7 275s
785279 295102 1.35785 43 88 1.35776 1.35803 0.02% 38.6 282s
794381 298468 1.35781 55 73 1.35776 1.35803 0.02% 38.6 285s
811152 304934 cutoff 68 1.35776 1.35803 0.02% 38.5 290s
827400 310874 1.35777 92 71 1.35776 1.35803 0.02% 38.4 295s
843531 316709 1.35776 55 70 1.35776 1.35803 0.02% 38.3 300s
859689 322375 cutoff 46 1.35776 1.35802 0.02% 38.2 305s
875647 327760 1.35779 46 69 1.35776 1.35802 0.02% 38.2 310s
891623 333304 1.35798 31 88 1.35776 1.35802 0.02% 38.1 315s
907752 338445 1.35797 53 82 1.35776 1.35802 0.02% 38.0 320s
923966 343848 1.35778 52 81 1.35776 1.35802 0.02% 38.0 325s
941617 349251 1.35794 50 74 1.35776 1.35802 0.02% 37.9 330s
955784 354198 1.35778 55 80 1.35776 1.35802 0.02% 37.8 335s
970053 359224 1.35780 42 83 1.35776 1.35802 0.02% 37.8 340s
984713 364694 1.35777 65 99 1.35776 1.35801 0.02% 37.7 345s
1002752 370960 1.35785 61 89 1.35776 1.35801 0.02% 37.6 350s
1015187 375638 cutoff 60 1.35776 1.35801 0.02% 37.6 355s
1029537 381053 1.35791 30 89 1.35776 1.35801 0.02% 37.5 360s
1045946 386868 1.35792 42 75 1.35776 1.35801 0.02% 37.5 365s
1062222 392497 1.35794 47 84 1.35776 1.35801 0.02% 37.4 370s
1077939 398053 cutoff 57 1.35776 1.35801 0.02% 37.3 375s
1093941 403782 1.35783 60 95 1.35776 1.35801 0.02% 37.3 380s
1107892 408465 1.35786 51 78 1.35776 1.35801 0.02% 37.2 385s
1124401 413863 1.35787 41 82 1.35776 1.35801 0.02% 37.2 390s
1138811 418586 1.35777 53 70 1.35776 1.35801 0.02% 37.1 395s
1153205 423280 cutoff 50 1.35776 1.35800 0.02% 37.1 400s
1169055 428424 cutoff 34 1.35776 1.35800 0.02% 37.1 405s
1178381 431537 1.35789 54 70 1.35776 1.35800 0.02% 37.0 410s
1195150 437870 1.35778 62 87 1.35776 1.35800 0.02% 37.0 415s
1211404 443297 1.35796 36 75 1.35776 1.35800 0.02% 36.9 420s
1225733 448005 1.35782 57 88 1.35776 1.35800 0.02% 36.9 425s
1241801 453264 cutoff 55 1.35776 1.35800 0.02% 36.8 430s
1252619 457038 1.35776 58 84 1.35776 1.35800 0.02% 36.8 435s
1268855 462062 1.35777 63 60 1.35776 1.35800 0.02% 36.7 440s
1279469 465724 1.35780 56 87 1.35776 1.35800 0.02% 36.7 445s
1295702 471466 1.35776 67 78 1.35776 1.35800 0.02% 36.7 450s
1312201 476739 1.35784 58 93 1.35776 1.35800 0.02% 36.6 455s
1328185 482563 1.35794 54 95 1.35776 1.35800 0.02% 36.6 460s
1344227 487855 1.35784 55 76 1.35776 1.35799 0.02% 36.5 465s
1360672 493354 1.35786 36 79 1.35776 1.35799 0.02% 36.5 470s
1376606 498829 1.35777 66 89 1.35776 1.35799 0.02% 36.4 475s
1390800 503592 cutoff 34 1.35776 1.35799 0.02% 36.4 480s
1405132 508318 cutoff 55 1.35776 1.35799 0.02% 36.3 485s
1421334 513862 1.35795 52 88 1.35776 1.35799 0.02% 36.3 490s
1435893 518601 cutoff 47 1.35776 1.35799 0.02% 36.3 495s
1446673 521991 cutoff 32 1.35776 1.35799 0.02% 36.2 500s
1462966 527248 cutoff 55 1.35776 1.35799 0.02% 36.2 505s
1479113 532630 cutoff 58 1.35776 1.35799 0.02% 36.1 510s
*1486494 534977 102 1.3577588 1.35799 0.02% 36.1 512s
1491247 536775 1.35793 51 73 1.35776 1.35799 0.02% 36.1 515s
1507988 542631 1.35778 55 75 1.35776 1.35799 0.02% 36.1 520s
1522496 547491 1.35787 50 72 1.35776 1.35799 0.02% 36.0 525s
1540383 553403 1.35784 56 74 1.35776 1.35799 0.02% 36.0 530s
1556531 558959 1.35782 53 81 1.35776 1.35799 0.02% 36.0 535s
1572589 563588 1.35791 46 77 1.35776 1.35798 0.02% 35.9 540s
1588483 568986 cutoff 51 1.35776 1.35798 0.02% 35.9 545s
1599099 572505 1.35778 67 76 1.35776 1.35798 0.02% 35.9 550s
*1607471 574488 104 1.3577590 1.35798 0.02% 35.8 552s
*1607483 574363 105 1.3577591 1.35798 0.02% 35.8 552s
1616180 577383 cutoff 42 1.35776 1.35798 0.02% 35.8 555s
1632110 582194 1.35776 53 61 1.35776 1.35798 0.02% 35.8 560s
1648142 587626 1.35787 55 79 1.35776 1.35798 0.02% 35.7 565s
1664127 592502 1.35776 58 89 1.35776 1.35798 0.02% 35.7 570s
1682190 598431 cutoff 60 1.35776 1.35798 0.02% 35.7 575s
1692938 601733 cutoff 49 1.35776 1.35798 0.02% 35.6 580s
1709339 606902 1.35783 57 90 1.35776 1.35798 0.02% 35.6 585s
1723499 611513 cutoff 76 1.35776 1.35798 0.02% 35.6 590s
1739719 616399 1.35781 56 81 1.35776 1.35798 0.02% 35.5 595s
1755658 621443 1.35778 57 78 1.35776 1.35798 0.02% 35.5 600s
1766048 625290 cutoff 92 1.35776 1.35798 0.02% 35.5 605s
1783274 631154 1.35778 63 91 1.35776 1.35798 0.02% 35.4 610s
1799505 636239 cutoff 56 1.35776 1.35798 0.02% 35.4 615s
1815568 641096 1.35792 48 61 1.35776 1.35798 0.02% 35.4 620s
1833507 647076 cutoff 58 1.35776 1.35797 0.02% 35.4 625s
1849517 652312 cutoff 56 1.35776 1.35797 0.02% 35.3 630s
1865741 657269 1.35779 51 80 1.35776 1.35797 0.02% 35.3 635s
1883498 662960 1.35791 52 77 1.35776 1.35797 0.02% 35.3 640s
1899683 668342 cutoff 45 1.35776 1.35797 0.02% 35.2 645s
1915461 673423 1.35782 55 82 1.35776 1.35797 0.02% 35.2 650s
1926254 676811 cutoff 58 1.35776 1.35797 0.02% 35.2 655s
*1936426 674876 106 1.3577601 1.35797 0.02% 35.2 658s
1943144 677329 1.35784 55 75 1.35776 1.35797 0.02% 35.1 660s
1959353 682760 1.35780 43 74 1.35776 1.35797 0.02% 35.1 665s
1975389 687655 cutoff 52 1.35776 1.35797 0.02% 35.1 670s
1993349 693618 cutoff 47 1.35776 1.35797 0.02% 35.1 675s
2009239 698264 1.35782 57 80 1.35776 1.35797 0.02% 35.0 680s
2025378 702835 1.35784 42 80 1.35776 1.35797 0.02% 35.0 685s
2035712 706115 1.35784 45 75 1.35776 1.35797 0.02% 35.0 690s
2052697 711039 1.35781 60 73 1.35776 1.35797 0.02% 35.0 695s
2070657 717383 1.35777 71 66 1.35776 1.35797 0.02% 34.9 700s
2087233 722561 1.35779 59 86 1.35776 1.35797 0.02% 34.9 705s
2104669 727931 1.35786 59 98 1.35776 1.35797 0.02% 34.9 710s
2120649 732783 1.35778 51 66 1.35776 1.35797 0.02% 34.8 715s
2136713 737330 1.35785 38 74 1.35776 1.35796 0.02% 34.8 720s
2154638 742958 cutoff 49 1.35776 1.35796 0.02% 34.8 725s
2167522 747136 1.35780 58 97 1.35776 1.35796 0.02% 34.8 730s
2183666 751899 cutoff 60 1.35776 1.35796 0.01% 34.7 735s
2199781 756805 1.35776 89 79 1.35776 1.35796 0.01% 34.7 740s
2210581 760320 1.35777 63 71 1.35776 1.35796 0.01% 34.7 745s
2226531 766018 cutoff 63 1.35776 1.35796 0.01% 34.7 750s
2243092 771208 1.35776 59 74 1.35776 1.35796 0.01% 34.6 755s
2259288 776303 1.35777 64 76 1.35776 1.35796 0.01% 34.6 760s
2275530 781370 cutoff 49 1.35776 1.35796 0.01% 34.6 765s
2291265 786348 cutoff 69 1.35776 1.35796 0.01% 34.5 770s
2303503 790240 1.35778 50 75 1.35776 1.35796 0.01% 34.5 775s
2318211 794845 cutoff 59 1.35776 1.35796 0.01% 34.5 780s
2334265 799330 cutoff 67 1.35776 1.35796 0.01% 34.5 785s
2350640 804060 1.35780 56 92 1.35776 1.35796 0.01% 34.5 790s
2366838 809643 cutoff 41 1.35776 1.35796 0.01% 34.4 795s
2383059 814875 1.35778 66 78 1.35776 1.35796 0.01% 34.4 800s
2396925 819227 1.35784 42 74 1.35776 1.35796 0.01% 34.4 805s
2414971 824553 cutoff 54 1.35776 1.35796 0.01% 34.4 810s
2429200 829193 cutoff 45 1.35776 1.35796 0.01% 34.3 815s
2443779 833505 cutoff 66 1.35776 1.35796 0.01% 34.3 820s
2457966 837944 1.35785 47 60 1.35776 1.35796 0.01% 34.3 825s
2474107 842890 1.35778 62 87 1.35776 1.35796 0.01% 34.3 830s
2490103 847739 1.35787 53 85 1.35776 1.35796 0.01% 34.3 835s
2501022 850878 1.35787 57 95 1.35776 1.35796 0.01% 34.2 840s
2517361 855999 1.35778 61 71 1.35776 1.35795 0.01% 34.2 845s
2533733 860618 1.35778 50 77 1.35776 1.35795 0.01% 34.2 850s
2549423 865489 1.35787 46 82 1.35776 1.35795 0.01% 34.2 855s
2563983 870094 1.35777 36 73 1.35776 1.35795 0.01% 34.2 860s
2576104 873647 1.35781 53 90 1.35776 1.35795 0.01% 34.1 865s
2592698 878440 1.35779 56 83 1.35776 1.35795 0.01% 34.1 870s
2608867 883213 1.35788 62 85 1.35776 1.35795 0.01% 34.1 875s
2623359 887564 1.35781 51 72 1.35776 1.35795 0.01% 34.1 880s
2639139 891935 cutoff 61 1.35776 1.35795 0.01% 34.1 885s
2649917 895146 1.35780 61 85 1.35776 1.35795 0.01% 34.1 890s
2666352 900020 cutoff 54 1.35776 1.35795 0.01% 34.0 895s
2680769 904080 1.35777 46 64 1.35776 1.35795 0.01% 34.0 900s
2697175 909157 cutoff 41 1.35776 1.35795 0.01% 34.0 905s
2713231 913485 1.35781 54 84 1.35776 1.35795 0.01% 34.0 910s
2727508 917955 cutoff 71 1.35776 1.35795 0.01% 34.0 915s
2740221 921808 cutoff 57 1.35776 1.35795 0.01% 33.9 920s
2756605 926485 1.35777 68 84 1.35776 1.35795 0.01% 33.9 925s
2772669 931295 1.35782 50 88 1.35776 1.35795 0.01% 33.9 930s
2790843 937414 1.35791 56 91 1.35776 1.35795 0.01% 33.9 935s
2806810 942295 1.35789 49 89 1.35776 1.35795 0.01% 33.8 940s
2822748 947320 1.35784 64 89 1.35776 1.35795 0.01% 33.8 945s
2835336 951171 cutoff 67 1.35776 1.35795 0.01% 33.8 950s
2851976 956766 cutoff 57 1.35776 1.35795 0.01% 33.8 955s
2867755 961421 cutoff 67 1.35776 1.35795 0.01% 33.8 960s
2880158 965196 cutoff 60 1.35776 1.35795 0.01% 33.8 965s
2894888 969326 1.35785 57 87 1.35776 1.35795 0.01% 33.7 970s
2912889 974690 cutoff 50 1.35776 1.35795 0.01% 33.7 975s
2928741 979008 cutoff 61 1.35776 1.35795 0.01% 33.7 980s
2944952 983542 1.35779 62 81 1.35776 1.35794 0.01% 33.7 985s
2961068 988060 1.35786 56 83 1.35776 1.35794 0.01% 33.7 990s
2977327 993309 1.35778 47 72 1.35776 1.35794 0.01% 33.6 995s
2991341 997181 1.35783 44 67 1.35776 1.35794 0.01% 33.6 1000s
3007754 1001973 cutoff 63 1.35776 1.35794 0.01% 33.6 1005s
3023703 1006658 1.35782 52 64 1.35776 1.35794 0.01% 33.6 1010s
3039774 1010898 cutoff 58 1.35776 1.35794 0.01% 33.6 1015s
3055703 1015215 1.35783 45 84 1.35776 1.35794 0.01% 33.6 1020s
3068546 1018770 1.35785 55 78 1.35776 1.35794 0.01% 33.5 1025s
3083072 1022691 1.35778 51 74 1.35776 1.35794 0.01% 33.5 1030s
3099230 1027283 1.35788 53 77 1.35776 1.35794 0.01% 33.5 1035s
3115512 1031762 1.35787 64 85 1.35776 1.35794 0.01% 33.5 1040s
3131349 1036424 cutoff 56 1.35776 1.35794 0.01% 33.5 1045s
3147423 1041140 1.35780 62 72 1.35776 1.35794 0.01% 33.5 1050s
3158219 1044001 1.35778 46 67 1.35776 1.35794 0.01% 33.5 1055s
3174382 1048858 1.35786 56 86 1.35776 1.35794 0.01% 33.4 1060s
3190552 1053263 1.35792 53 87 1.35776 1.35794 0.01% 33.4 1065s
3206703 1057552 1.35781 60 76 1.35776 1.35794 0.01% 33.4 1070s
3221271 1061979 1.35793 51 78 1.35776 1.35794 0.01% 33.4 1075s
3237429 1066804 1.35783 57 78 1.35776 1.35794 0.01% 33.4 1080s
3248191 1070031 1.35780 68 95 1.35776 1.35794 0.01% 33.4 1085s
3264290 1074847 1.35785 54 91 1.35776 1.35794 0.01% 33.4 1090s
3280434 1079474 1.35776 64 79 1.35776 1.35794 0.01% 33.3 1095s
3296789 1084356 cutoff 34 1.35776 1.35794 0.01% 33.3 1100s
3312722 1089159 cutoff 64 1.35776 1.35794 0.01% 33.3 1105s
3323429 1092332 1.35784 61 91 1.35776 1.35794 0.01% 33.3 1110s
3339987 1097070 1.35780 62 75 1.35776 1.35794 0.01% 33.3 1115s
3356096 1101849 1.35785 61 96 1.35776 1.35794 0.01% 33.3 1120s
3370776 1106242 cutoff 57 1.35776 1.35794 0.01% 33.2 1125s
3383286 1109784 cutoff 70 1.35776 1.35794 0.01% 33.2 1130s
3399455 1114454 cutoff 40 1.35776 1.35794 0.01% 33.2 1135s
3415350 1118956 1.35792 50 74 1.35776 1.35794 0.01% 33.2 1140s
3431462 1123942 1.35781 51 76 1.35776 1.35794 0.01% 33.2 1145s
3441788 1127109 1.35777 57 82 1.35776 1.35794 0.01% 33.2 1150s
3458767 1132207 1.35778 69 89 1.35776 1.35793 0.01% 33.2 1155s
3474992 1137198 1.35784 56 83 1.35776 1.35793 0.01% 33.1 1160s
3491024 1142458 1.35788 60 92 1.35776 1.35793 0.01% 33.1 1165s
3507023 1146815 1.35777 57 89 1.35776 1.35793 0.01% 33.1 1170s
3523120 1151251 1.35790 35 84 1.35776 1.35793 0.01% 33.1 1175s
3539202 1156126 1.35779 51 79 1.35776 1.35793 0.01% 33.1 1180s
3555233 1160560 cutoff 50 1.35776 1.35793 0.01% 33.1 1185s
3571224 1165017 1.35786 58 74 1.35776 1.35793 0.01% 33.0 1190s
3589000 1170279 1.35782 52 77 1.35776 1.35793 0.01% 33.0 1195s
3605144 1174973 1.35781 68 89 1.35776 1.35793 0.01% 33.0 1200s
3615901 1177994 cutoff 46 1.35776 1.35793 0.01% 33.0 1205s
3632345 1182261 1.35783 47 81 1.35776 1.35793 0.01% 33.0 1210s
3648441 1186660 cutoff 54 1.35776 1.35793 0.01% 33.0 1215s
3664661 1190574 1.35786 52 79 1.35776 1.35793 0.01% 33.0 1220s
3680692 1194985 1.35791 56 93 1.35776 1.35793 0.01% 32.9 1225s
3694730 1198510 1.35785 45 68 1.35776 1.35793 0.01% 32.9 1230s
3705618 1201499 1.35779 60 94 1.35776 1.35793 0.01% 32.9 1235s
3721985 1205818 1.35778 45 80 1.35776 1.35793 0.01% 32.9 1240s
3738439 1211019 1.35784 53 91 1.35776 1.35793 0.01% 32.9 1245s
3754513 1215307 1.35780 62 76 1.35776 1.35793 0.01% 32.9 1250s
3770453 1219376 cutoff 59 1.35776 1.35793 0.01% 32.9 1255s
3786547 1223768 1.35781 63 81 1.35776 1.35793 0.01% 32.9 1260s
3799075 1227169 1.35790 51 72 1.35776 1.35793 0.01% 32.9 1265s
3813658 1231353 1.35776 61 82 1.35776 1.35793 0.01% 32.8 1270s
3829990 1236616 1.35780 34 64 1.35776 1.35793 0.01% 32.8 1275s
3845903 1241013 cutoff 65 1.35776 1.35793 0.01% 32.8 1280s
3861871 1245306 1.35777 61 82 1.35776 1.35793 0.01% 32.8 1285s
3876492 1249175 cutoff 43 1.35776 1.35793 0.01% 32.8 1291s
3888872 1252609 1.35777 61 87 1.35776 1.35793 0.01% 32.8 1295s
3903630 1256501 1.35785 53 69 1.35776 1.35793 0.01% 32.8 1300s
3919919 1261269 1.35782 60 86 1.35776 1.35793 0.01% 32.8 1305s
3936134 1265693 1.35786 55 68 1.35776 1.35793 0.01% 32.7 1311s
3946563 1268663 1.35776 57 81 1.35776 1.35793 0.01% 32.7 1315s
3962804 1273169 cutoff 66 1.35776 1.35793 0.01% 32.7 1320s
3978833 1277674 cutoff 66 1.35776 1.35793 0.01% 32.7 1325s
3995043 1282459 1.35777 60 92 1.35776 1.35793 0.01% 32.7 1330s
4009203 1286647 1.35779 67 86 1.35776 1.35793 0.01% 32.7 1336s
4020046 1290181 1.35777 58 89 1.35776 1.35793 0.01% 32.7 1340s
4036739 1294816 cutoff 59 1.35776 1.35793 0.01% 32.7 1345s
4050865 1298797 1.35777 65 76 1.35776 1.35793 0.01% 32.7 1350s
4066940 1303481 1.35781 49 74 1.35776 1.35793 0.01% 32.6 1355s
4083084 1308022 cutoff 47 1.35776 1.35793 0.01% 32.6 1360s
4099178 1312451 cutoff 57 1.35776 1.35792 0.01% 32.6 1365s
4111636 1316109 1.35777 59 74 1.35776 1.35792 0.01% 32.6 1370s
4126069 1319984 1.35790 43 85 1.35776 1.35792 0.01% 32.6 1375s
4142219 1324600 1.35777 49 79 1.35776 1.35792 0.01% 32.6 1380s
4156475 1329080 cutoff 64 1.35776 1.35792 0.01% 32.6 1385s
4167158 1331733 1.35777 60 74 1.35776 1.35792 0.01% 32.6 1390s
4183503 1336038 1.35776 59 76 1.35776 1.35792 0.01% 32.6 1395s
4199620 1340156 1.35780 54 76 1.35776 1.35792 0.01% 32.5 1400s
4214064 1344183 cutoff 50 1.35776 1.35792 0.01% 32.5 1406s
4224907 1347337 cutoff 59 1.35776 1.35792 0.01% 32.5 1410s
4241167 1351779 1.35787 59 88 1.35776 1.35792 0.01% 32.5 1415s
4257288 1356218 cutoff 57 1.35776 1.35792 0.01% 32.5 1420s
4271837 1360428 1.35777 61 62 1.35776 1.35792 0.01% 32.5 1425s
4284020 1364051 cutoff 61 1.35776 1.35792 0.01% 32.5 1430s
4298558 1368550 1.35782 62 90 1.35776 1.35792 0.01% 32.5 1435s
4314856 1372522 1.35777 63 79 1.35776 1.35792 0.01% 32.5 1440s
4329066 1376264 1.35781 50 66 1.35776 1.35792 0.01% 32.4 1445s
4344910 1380772 1.35789 62 93 1.35776 1.35792 0.01% 32.4 1450s
4359275 1384175 1.35785 60 82 1.35776 1.35792 0.01% 32.4 1455s
4375547 1389346 cutoff 58 1.35776 1.35792 0.01% 32.4 1460s
4386142 1392506 1.35784 54 87 1.35776 1.35792 0.01% 32.4 1465s
4402722 1396898 1.35782 51 59 1.35776 1.35792 0.01% 32.4 1470s
4417124 1400981 1.35788 59 94 1.35776 1.35792 0.01% 32.4 1475s
4433002 1405247 1.35780 64 69 1.35776 1.35792 0.01% 32.4 1480s
4445535 1408849 1.35779 51 85 1.35776 1.35792 0.01% 32.4 1485s
4460076 1413053 1.35787 52 74 1.35776 1.35792 0.01% 32.4 1490s
4474488 1416876 1.35777 75 101 1.35776 1.35792 0.01% 32.3 1495s
4490727 1421450 1.35776 59 74 1.35776 1.35792 0.01% 32.3 1500s
4502659 1425189 1.35777 62 79 1.35776 1.35792 0.01% 32.3 1505s
4517668 1429638 1.35782 60 74 1.35776 1.35792 0.01% 32.3 1510s
4533870 1434533 cutoff 42 1.35776 1.35792 0.01% 32.3 1515s
4548169 1438514 1.35778 67 86 1.35776 1.35792 0.01% 32.3 1520s
4561882 1442287 cutoff 65 1.35776 1.35792 0.01% 32.3 1525s
4576695 1446467 1.35779 68 87 1.35776 1.35792 0.01% 32.3 1530s
4586858 1449184 1.35788 58 82 1.35776 1.35792 0.01% 32.3 1535s
4601925 1453572 1.35780 50 72 1.35776 1.35792 0.01% 32.2 1540s
4616428 1457599 1.35776 63 76 1.35776 1.35792 0.01% 32.2 1545s
4632430 1461633 cutoff 52 1.35776 1.35792 0.01% 32.2 1550s
4646949 1465437 1.35779 55 90 1.35776 1.35792 0.01% 32.2 1555s
4662911 1469301 cutoff 59 1.35776 1.35792 0.01% 32.2 1560s
4675349 1472406 cutoff 68 1.35776 1.35792 0.01% 32.2 1565s
4687815 1475736 1.35787 59 89 1.35776 1.35792 0.01% 32.2 1570s
4704279 1480489 1.35777 56 68 1.35776 1.35792 0.01% 32.2 1575s
4718589 1484499 1.35783 61 94 1.35776 1.35792 0.01% 32.2 1580s
4734624 1488803 1.35779 63 77 1.35776 1.35792 0.01% 32.2 1585s
4747133 1491955 1.35786 53 67 1.35776 1.35792 0.01% 32.1 1591s
4758027 1495223 1.35777 63 48 1.35776 1.35792 0.01% 32.1 1595s
4772693 1499524 1.35779 64 78 1.35776 1.35792 0.01% 32.1 1600s
4788782 1503713 cutoff 63 1.35776 1.35792 0.01% 32.1 1605s
4803027 1507529 1.35788 59 92 1.35776 1.35792 0.01% 32.1 1610s
4817511 1511813 1.35776 58 76 1.35776 1.35792 0.01% 32.1 1615s
4827863 1514765 1.35789 52 75 1.35776 1.35792 0.01% 32.1 1620s
4842995 1518809 cutoff 53 1.35776 1.35792 0.01% 32.1 1625s
4857562 1522759 1.35780 58 76 1.35776 1.35791 0.01% 32.1 1630s
4873637 1527274 1.35791 60 84 1.35776 1.35791 0.01% 32.1 1635s
4889468 1531337 1.35777 58 103 1.35776 1.35791 0.01% 32.1 1640s
4903901 1535521 1.35790 62 85 1.35776 1.35791 0.01% 32.0 1645s
4918124 1539546 1.35782 64 98 1.35776 1.35791 0.01% 32.0 1650s
4934354 1544201 cutoff 55 1.35776 1.35791 0.01% 32.0 1655s
4950314 1548391 1.35778 70 79 1.35776 1.35791 0.01% 32.0 1660s
4964785 1552547 1.35777 65 88 1.35776 1.35791 0.01% 32.0 1665s
4976928 1555977 1.35790 57 93 1.35776 1.35791 0.01% 32.0 1670s
4991570 1559971 cutoff 62 1.35776 1.35791 0.01% 32.0 1675s
5007670 1564338 cutoff 59 1.35776 1.35791 0.01% 32.0 1680s
5018416 1567042 cutoff 62 1.35776 1.35791 0.01% 32.0 1685s
5033033 1570724 1.35779 63 98 1.35776 1.35791 0.01% 32.0 1690s
5047370 1574227 1.35778 66 83 1.35776 1.35791 0.01% 31.9 1695s
5063441 1578723 1.35784 61 93 1.35776 1.35791 0.01% 31.9 1700s
5077728 1582389 1.35789 59 91 1.35776 1.35791 0.01% 31.9 1705s
5092088 1586479 1.35786 53 90 1.35776 1.35791 0.01% 31.9 1710s
5104682 1589748 1.35780 52 77 1.35776 1.35791 0.01% 31.9 1715s
5118992 1593468 1.35784 43 85 1.35776 1.35791 0.01% 31.9 1720s
5135161 1597789 cutoff 52 1.35776 1.35791 0.01% 31.9 1725s
5145744 1600606 1.35781 42 71 1.35776 1.35791 0.01% 31.9 1730s
5160588 1604502 1.35777 63 69 1.35776 1.35791 0.01% 31.9 1735s
5175132 1608669 1.35780 60 78 1.35776 1.35791 0.01% 31.9 1740s
5190826 1612679 1.35779 55 63 1.35776 1.35791 0.01% 31.9 1745s
5203119 1616036 cutoff 54 1.35776 1.35791 0.01% 31.8 1750s
5217945 1620097 cutoff 49 1.35776 1.35791 0.01% 31.8 1755s
5232211 1623850 1.35781 62 92 1.35776 1.35791 0.01% 31.8 1760s
5248478 1628581 1.35785 62 86 1.35776 1.35791 0.01% 31.8 1765s
5262383 1632770 cutoff 60 1.35776 1.35791 0.01% 31.8 1770s
5271344 1635542 1.35781 55 89 1.35776 1.35791 0.01% 31.8 1775s
5286179 1639916 cutoff 60 1.35776 1.35791 0.01% 31.8 1780s
5302338 1644566 1.35787 61 92 1.35776 1.35791 0.01% 31.8 1785s
5316711 1648326 1.35778 57 94 1.35776 1.35791 0.01% 31.8 1790s
5330786 1651948 1.35786 58 88 1.35776 1.35791 0.01% 31.8 1795s
5346345 1656396 cutoff 58 1.35776 1.35791 0.01% 31.8 1800s
5361005 1660065 1.35777 55 64 1.35776 1.35791 0.01% 31.7 1805s
5375484 1664489 cutoff 52 1.35776 1.35791 0.01% 31.7 1810s
5387847 1667921 cutoff 57 1.35776 1.35791 0.01% 31.7 1815s
5402062 1671579 1.35786 52 82 1.35776 1.35791 0.01% 31.7 1820s
5414524 1675094 1.35782 67 94 1.35776 1.35791 0.01% 31.7 1825s
5427616 1678301 cutoff 54 1.35776 1.35791 0.01% 31.7 1830s
5443611 1682460 1.35782 49 72 1.35776 1.35791 0.01% 31.7 1835s
5457987 1686293 1.35782 62 96 1.35776 1.35791 0.01% 31.7 1840s
5468505 1688972 1.35781 51 83 1.35776 1.35791 0.01% 31.7 1845s
5483318 1692518 cutoff 54 1.35776 1.35791 0.01% 31.7 1850s
5497835 1696483 1.35777 63 77 1.35776 1.35791 0.01% 31.7 1855s
5512087 1700417 1.35779 44 82 1.35776 1.35791 0.01% 31.7 1860s
5528135 1704585 1.35778 44 83 1.35776 1.35791 0.01% 31.7 1866s
5538986 1707635 cutoff 45 1.35776 1.35791 0.01% 31.6 1870s
5553651 1711768 cutoff 46 1.35776 1.35791 0.01% 31.6 1875s
5568101 1715453 cutoff 39 1.35776 1.35791 0.01% 31.6 1880s
5582092 1719095 cutoff 52 1.35776 1.35791 0.01% 31.6 1885s
5596218 1722623 cutoff 59 1.35776 1.35791 0.01% 31.6 1890s
5606986 1725316 1.35778 53 93 1.35776 1.35791 0.01% 31.6 1895s
5621378 1729327 1.35781 50 73 1.35776 1.35791 0.01% 31.6 1900s
5635842 1733246 cutoff 67 1.35776 1.35791 0.01% 31.6 1905s
5650319 1737242 1.35778 68 86 1.35776 1.35791 0.01% 31.6 1910s
5664033 1740870 1.35783 60 87 1.35776 1.35791 0.01% 31.6 1915s
5678376 1744230 1.35786 55 83 1.35776 1.35791 0.01% 31.6 1920s
5689260 1747102 cutoff 61 1.35776 1.35791 0.01% 31.6 1925s
5703996 1751408 1.35780 64 86 1.35776 1.35791 0.01% 31.5 1930s
5718437 1754884 1.35776 61 70 1.35776 1.35791 0.01% 31.5 1935s
5732880 1759245 cutoff 57 1.35776 1.35791 0.01% 31.5 1940s
5743364 1761844 1.35776 57 82 1.35776 1.35791 0.01% 31.5 1945s
5756036 1765508 1.35779 68 85 1.35776 1.35791 0.01% 31.5 1950s
5770492 1769240 1.35784 63 91 1.35776 1.35791 0.01% 31.5 1955s
5783023 1772299 cutoff 54 1.35776 1.35791 0.01% 31.5 1960s
5797390 1776173 1.35783 51 90 1.35776 1.35790 0.01% 31.5 1965s
5811528 1779719 cutoff 54 1.35776 1.35790 0.01% 31.5 1970s
5822046 1782477 1.35778 64 82 1.35776 1.35790 0.01% 31.5 1975s
5837033 1786223 cutoff 56 1.35776 1.35790 0.01% 31.5 1980s
5849714 1790086 cutoff 59 1.35776 1.35790 0.01% 31.5 1985s
5864048 1793992 1.35779 66 83 1.35776 1.35790 0.01% 31.5 1990s
5878466 1798088 1.35782 46 75 1.35776 1.35790 0.01% 31.4 1995s
5887167 1800409 cutoff 61 1.35776 1.35790 0.01% 31.4 2000s
5902066 1804464 1.35777 53 74 1.35776 1.35790 0.01% 31.4 2005s
5916413 1808510 1.35780 67 92 1.35776 1.35790 0.01% 31.4 2010s
5930887 1812173 1.35785 48 72 1.35776 1.35790 0.01% 31.4 2015s
5945073 1815750 cutoff 55 1.35776 1.35790 0.01% 31.4 2020s
5959557 1819696 1.35783 60 88 1.35776 1.35790 0.01% 31.4 2025s
5973736 1823086 1.35783 49 79 1.35776 1.35790 0.01% 31.4 2030s
5989793 1827503 1.35776 71 83 1.35776 1.35790 0.01% 31.4 2035s
6004275 1831497 1.35777 55 91 1.35776 1.35790 0.01% 31.4 2040s
6016557 1834665 1.35786 60 88 1.35776 1.35790 0.01% 31.4 2045s
6031022 1838503 1.35786 52 82 1.35776 1.35790 0.01% 31.4 2050s
6039999 1840620 1.35778 60 91 1.35776 1.35790 0.01% 31.4 2055s
6054865 1844622 cutoff 65 1.35776 1.35790 0.01% 31.3 2060s
6069237 1848203 1.35784 65 87 1.35776 1.35790 0.01% 31.3 2065s
6083824 1852288 1.35783 59 93 1.35776 1.35790 0.01% 31.3 2070s
6098069 1856039 1.35778 68 86 1.35776 1.35790 0.01% 31.3 2075s
6112243 1859680 1.35779 51 74 1.35776 1.35790 0.01% 31.3 2080s
6126821 1863915 1.35782 59 87 1.35776 1.35790 0.01% 31.3 2085s
6141205 1867978 1.35786 51 76 1.35776 1.35790 0.01% 31.3 2090s
6155372 1871549 1.35781 57 65 1.35776 1.35790 0.01% 31.3 2095s
6169465 1874791 cutoff 63 1.35776 1.35790 0.01% 31.3 2100s
6183878 1878567 cutoff 62 1.35776 1.35790 0.01% 31.3 2105s
6197823 1882134 cutoff 57 1.35776 1.35790 0.01% 31.3 2110s
6210164 1885283 1.35783 61 83 1.35776 1.35790 0.01% 31.3 2115s
6224891 1888973 1.35778 59 75 1.35776 1.35790 0.01% 31.3 2120s
6239076 1892533 1.35777 63 82 1.35776 1.35790 0.01% 31.2 2125s
6252034 1895940 1.35787 55 79 1.35776 1.35790 0.01% 31.2 2131s
6264436 1899478 1.35780 64 81 1.35776 1.35790 0.01% 31.2 2135s
6276881 1902532 cutoff 65 1.35776 1.35790 0.01% 31.2 2140s
6291471 1906300 1.35777 53 73 1.35776 1.35790 0.01% 31.2 2145s
6305809 1909856 cutoff 53 1.35776 1.35790 0.01% 31.2 2150s
6320063 1914089 cutoff 71 1.35776 1.35790 0.01% 31.2 2155s
6330760 1916591 1.35776 67 81 1.35776 1.35790 0.01% 31.2 2161s
6341769 1919815 1.35784 54 77 1.35776 1.35790 0.01% 31.2 2165s
6356179 1923713 1.35780 56 94 1.35776 1.35790 0.01% 31.2 2170s
6370815 1927613 1.35781 54 88 1.35776 1.35790 0.01% 31.2 2175s
6385133 1931708 1.35779 78 110 1.35776 1.35790 0.01% 31.2 2180s
6399300 1935402 1.35780 53 67 1.35776 1.35790 0.01% 31.2 2185s
6413597 1939334 1.35777 51 76 1.35776 1.35790 0.01% 31.2 2190s
6425938 1942696 cutoff 69 1.35776 1.35790 0.01% 31.1 2195s
6440295 1946434 1.35781 61 93 1.35776 1.35790 0.01% 31.1 2200s
6454594 1950204 cutoff 70 1.35776 1.35790 0.01% 31.1 2205s
6468818 1953978 1.35778 62 87 1.35776 1.35790 0.01% 31.1 2210s
6483092 1957516 1.35779 47 63 1.35776 1.35790 0.01% 31.1 2215s
6492058 1959850 cutoff 65 1.35776 1.35790 0.01% 31.1 2220s
6506977 1963708 1.35782 65 79 1.35776 1.35790 0.01% 31.1 2225s
6519677 1967055 1.35785 42 70 1.35776 1.35790 0.01% 31.1 2230s
6533990 1971074 1.35784 54 86 1.35776 1.35790 0.01% 31.1 2235s
6548254 1974583 1.35778 72 89 1.35776 1.35790 0.01% 31.1 2240s
6562633 1978462 1.35782 51 73 1.35776 1.35790 0.01% 31.1 2245s
6576839 1982198 1.35779 57 88 1.35776 1.35790 0.01% 31.1 2250s
6591054 1985876 1.35780 41 77 1.35776 1.35790 0.01% 31.1 2255s
6605301 1989179 cutoff 60 1.35776 1.35790 0.01% 31.0 2260s
6617857 1992129 1.35787 50 79 1.35776 1.35790 0.01% 31.0 2266s
6626904 1995089 cutoff 71 1.35776 1.35790 0.01% 31.0 2270s
6641540 1998927 1.35784 61 101 1.35776 1.35790 0.01% 31.0 2275s
6655887 2002762 cutoff 70 1.35776 1.35790 0.01% 31.0 2280s
6668392 2006236 1.35783 58 70 1.35776 1.35790 0.01% 31.0 2285s
6682852 2010061 cutoff 63 1.35776 1.35790 0.01% 31.0 2290s
6697069 2013491 cutoff 50 1.35776 1.35790 0.01% 31.0 2295s
6709541 2016710 cutoff 66 1.35776 1.35790 0.01% 31.0 2300s
6723721 2020180 cutoff 68 1.35776 1.35790 0.01% 31.0 2305s
6738138 2023848 1.35785 56 88 1.35776 1.35790 0.01% 31.0 2310s
6752323 2027497 cutoff 65 1.35776 1.35790 0.01% 31.0 2315s
6765092 2030845 1.35787 52 87 1.35776 1.35790 0.01% 31.0 2320s
6778839 2034591 1.35778 62 86 1.35776 1.35790 0.01% 31.0 2325s
6791544 2037998 cutoff 58 1.35776 1.35790 0.01% 31.0 2330s
6803937 2041489 1.35784 62 85 1.35776 1.35790 0.01% 30.9 2335s
6818543 2045429 1.35788 53 91 1.35776 1.35790 0.01% 30.9 2340s
6831214 2048687 1.35781 64 88 1.35776 1.35790 0.01% 30.9 2345s
6845498 2052533 1.35779 63 77 1.35776 1.35790 0.01% 30.9 2350s
6852161 2054189 cutoff 64 1.35776 1.35790 0.01% 30.9 2355s
Cutting planes:
PSD: 621
Explored 6854080 nodes (211940727 simplex iterations) in 2355.21 seconds
Thread count was 12 (of 12 available processors)
Solution count 10: 1.35776 1.35776 1.35776 ... 1.35774
Optimal solution found (tolerance 1.00e-04)
Warning: max constraint violation (6.6210e-06) exceeds tolerance
Best objective 1.357764008805e+00, best bound 1.357895859649e+00, gap 0.0097%
Execution time: 2357.12 secondsI do not send the log using 20 variables because I interrupted that execution after 12 hours before it was finished and it is incomplete.
Best regards,
Álvaro
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Hi Álvaro,
Could you try providing finite bounds for your \(\texttt{weights}\) variables? The bounds of your \(\texttt{weights}\) variables are already limited to \([-1,1]\) due to the equality constraint, but maybe you can provide tighter bounds, e.g., make all (or a subset of the variables) positive. It is best to provide as tight bounds as possible. This is essential to make the fast solution of nonconvex problems possible.
You could also provide LP files for the cases with 11-15 variables such that the Community has a chance to have a closer look. For that, please have a look at Posting to the Community Forum.
Best regards,
Jaromił0 -
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Hello Jaromil,
Thank you for your answers. The only bound I know for the weights is the [-1, 1] interval, but after the problem is solved I do not have any prior knowledge of other bounds. Here I provide a link for downloading the LP files for the cases with 11 up to 16 variables: https://we.tl/t-dfJ3Pb4NNx
Best,
Álvaro
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Hi Álvaro,
The files contain LOGs only.
Maybe, before providing the LP files, could you try to reformulate your PCA problem in a convex way? The stacksexchange post Is PCA optimization convex? is a great guide for the reformulation and contains a lot of interesting details.
Best regards,
Jaromił0 -
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