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I encountered the problem that QP solver works in an unstable way. I am solving the QP problems with the same structure at every iteration. The objective function is the same quadratic term, and each iteration I add one extra constraint to the model. In the first 23 iterations the QP solver works ok. However, the QP solver suddenly takes much more time to solve after some iterations.

For example at iteration 23, the log is like:

Gurobi Optimizer version 9.1.1 build v9.1.1rc0 (mac64)
Thread count: 8 physical cores, 16 logical processors, using up to 16 threads
Optimize a model with 164054 rows, 107382 columns and 1338753 nonzeros
Model fingerprint: 0xc2fda6a2
Model has 1876 quadratic objective terms
Coefficient statistics:
Matrix range [3e-08, 9e+04]
Objective range [4e-11, 2e+01]
QObjective range [7e-02, 2e+00]
Bounds range [6e-01, 3e+05]
RHS range [1e-01, 2e+05]
Warning: Model contains large matrix coefficient range
Consider reformulating model or setting NumericFocus parameter
to avoid numerical issues.
Presolve removed 109551 rows and 60994 columns
Presolve time: 0.45s
Presolved: 54503 rows, 46388 columns, 789446 nonzeros
Presolved model has 1876 quadratic objective terms
Ordering time: 0.53s
Ordering time: 1.20s

Barrier statistics:
Dense cols : 62
AA' NZ : 1.796e+06
Factor NZ : 1.611e+07 (roughly 170 MBytes of memory)
Factor Ops : 6.055e+09 (less than 1 second per iteration)
Threads : 8

Objective Residual
Iter Primal Dual Primal Dual Compl Time
0 6.48426994e+13 -1.95781010e+14 3.53e+07 2.39e+05 7.83e+10 2s
1 2.30780707e+13 -1.48965948e+14 1.91e+07 1.30e+05 4.56e+10 2s
2 9.17891136e+12 -1.33874808e+14 1.15e+07 7.82e+04 3.07e+10 2s
3 7.91137041e+11 -1.10030794e+14 3.51e+06 2.38e+04 1.15e+10 3s
4 2.17837548e+11 -8.89072421e+13 1.93e+06 1.31e+04 6.78e+09 3s
5 6.03769303e+10 -7.85266738e+13 1.04e+06 7.03e+03 4.37e+09 3s
6 3.61265994e+09 -4.91634200e+13 2.42e+05 1.64e+03 1.36e+09 3s
7 1.22959900e+09 -3.84134934e+13 1.33e+05 9.01e+02 9.15e+08 3s
8 6.77854454e+08 -3.02708227e+13 9.64e+04 6.54e+02 7.27e+08 3s
9 2.54139180e+08 -2.18700152e+13 5.69e+04 3.86e+02 5.12e+08 3s
10 1.67321977e+08 -1.87392597e+13 4.53e+04 3.08e+02 4.32e+08 4s
11 6.48426994e+13 -1.95781010e+14 3.53e+07 2.39e+05 7.83e+10 4s
12 2.99831742e+11 -3.34320097e+12 2.22e+06 1.51e+04 5.52e+08 4s
13 2.43309692e+10 -8.10105087e+11 5.90e+05 3.93e+03 6.64e+07 4s
14 1.54235357e+09 -1.78797505e+11 1.41e+05 9.36e+02 7.90e+06 4s
15 2.23435394e+08 -6.80312249e+10 5.36e+04 3.57e+02 2.24e+06 4s
16 1.05231586e+08 -4.58000748e+10 3.61e+04 2.40e+02 1.35e+06 4s
17 5.33354313e+07 -3.08732269e+10 2.43e+04 1.62e+02 8.34e+05 5s
18 2.23223875e+07 -1.88637387e+10 1.48e+04 9.88e+01 4.80e+05 5s
19 2.37706916e+06 -4.87406691e+09 3.84e+03 2.56e+01 1.08e+05 5s
20 1.24593691e+06 -3.39225923e+09 2.67e+03 1.78e+01 6.95e+04 5s
21 4.55461665e+05 -1.86016525e+09 1.46e+03 9.74e+00 3.34e+04 5s
22 2.75024024e+05 -1.34176297e+09 1.06e+03 7.03e+00 2.26e+04 5s
23 1.52878349e+05 -9.07675288e+08 7.13e+02 4.75e+00 1.40e+04 6s
24 8.06500345e+04 -5.56255713e+08 4.37e+02 2.91e+00 7.81e+03 6s
25 5.77189648e+04 -4.10229971e+08 3.22e+02 2.15e+00 5.48e+03 6s
26 5.16328832e+04 -3.66510614e+08 2.88e+02 1.92e+00 4.82e+03 6s
27 3.47356116e+04 -2.36157205e+08 1.85e+02 1.23e+00 2.95e+03 6s
28 3.01151007e+04 -1.90166354e+08 1.49e+02 9.94e-01 2.31e+03 6s
29 2.23818033e+04 -1.10377007e+08 8.66e+01 5.77e-01 1.27e+03 7s
30 1.85170060e+04 -6.42490949e+07 5.04e+01 3.36e-01 7.12e+02 7s
31 1.71181663e+04 -4.73409797e+07 3.72e+01 2.47e-01 5.15e+02 7s
32 1.49475469e+04 -1.68368396e+07 1.32e+01 8.79e-02 1.76e+02 7s
33 1.31215513e+04 -2.05208999e+06 1.61e+00 1.07e-02 2.11e+01 7s
34 1.00292399e+04 -2.76420971e+05 2.27e-01 1.51e-03 2.91e+00 7s
35 8.53543189e+03 -1.95526428e+05 1.64e-01 1.09e-03 2.08e+00 7s
36 8.17136737e+03 -1.72343632e+05 1.46e-01 9.69e-04 1.84e+00 8s
37 7.89546501e+03 -1.60723517e+05 1.36e-01 9.08e-04 1.71e+00 8s
38 7.02957960e+03 -1.32061015e+05 1.14e-01 7.57e-04 1.41e+00 8s
39 6.36319538e+03 -1.12625997e+05 9.83e-02 6.54e-04 1.21e+00 8s
40 5.94751134e+03 -1.01019910e+05 8.90e-02 5.92e-04 1.09e+00 8s
41 4.89409241e+03 -8.14138282e+04 7.26e-02 4.83e-04 8.77e-01 8s
42 4.82765154e+03 -7.95756578e+04 7.11e-02 4.73e-04 8.57e-01 8s
43 4.49253510e+03 -7.09301469e+04 6.40e-02 4.26e-04 7.66e-01 9s
44 4.06926048e+03 -6.20694963e+04 5.67e-02 3.77e-04 6.72e-01 9s
45 2.91914045e+03 -3.86666099e+04 3.70e-02 2.46e-04 4.22e-01 9s
46 2.33954512e+03 -2.79061760e+04 2.75e-02 1.83e-04 3.07e-01 9s
47 1.73989926e+03 -1.77681331e+04 1.83e-02 1.22e-04 1.98e-01 9s
48 1.10610644e+03 -1.06390454e+04 1.17e-02 7.76e-05 1.19e-01 9s
49 8.24671446e+02 -7.44495159e+03 8.67e-03 5.77e-05 8.38e-02 10s
50 5.15422456e+02 -4.64783175e+03 5.67e-03 3.78e-05 5.23e-02 10s
51 2.97241338e+02 -2.69311150e+03 3.49e-03 2.32e-05 3.02e-02 10s
52 1.98086373e+02 -1.85105945e+03 2.47e-03 1.64e-05 2.07e-02 10s
53 1.41178629e+02 -1.38133525e+03 1.87e-03 1.25e-05 1.54e-02 10s
54 8.30805446e+01 -8.87014487e+02 1.22e-03 8.15e-06 9.79e-03 10s
55 5.47577519e+01 -5.98521969e+02 8.85e-04 5.89e-06 6.59e-03 10s
56 3.78199633e+01 -4.51993998e+02 6.48e-04 4.31e-06 4.94e-03 11s
57 2.85885779e+01 -3.84966836e+02 4.79e-04 3.19e-06 4.16e-03 11s
58 2.27839752e+01 -3.12861197e+02 4.00e-04 2.66e-06 3.37e-03 11s
59 1.63470815e+01 -2.37402301e+02 2.75e-04 1.83e-06 2.55e-03 11s
60 1.22887280e+01 -1.86459669e+02 2.07e-04 1.37e-06 1.99e-03 11s
61 1.11098501e+01 -1.70371677e+02 1.87e-04 1.24e-06 1.82e-03 11s
62 8.93789330e+00 -1.38729998e+02 1.44e-04 9.61e-07 1.48e-03 12s
63 7.41122958e+00 -1.16533131e+02 1.11e-04 7.39e-07 1.24e-03 12s
64 6.93434914e+00 -1.09336253e+02 1.05e-04 6.96e-07 1.16e-03 12s
65 6.78537877e+00 -1.07037751e+02 1.02e-04 6.82e-07 1.14e-03 12s
66 5.82949231e+00 -9.23652800e+01 8.46e-05 5.63e-07 9.82e-04 12s
67 5.06515293e+00 -7.95160040e+01 7.09e-05 4.72e-07 8.45e-04 12s
68 4.61638871e+00 -7.16159092e+01 6.42e-05 4.27e-07 7.62e-04 12s
69 4.13202163e+00 -6.43020115e+01 5.57e-05 3.71e-07 6.84e-04 13s
70 2.60266838e+00 -3.96274408e+01 3.23e-05 2.15e-07 4.22e-04 13s
71 2.22283189e+00 -3.37367968e+01 2.31e-05 1.54e-07 3.59e-04 13s
72 1.65543347e+00 -2.41542930e+01 1.13e-05 7.50e-08 2.57e-04 13s
73 1.36468987e+00 -1.81225617e+01 7.85e-06 5.22e-08 1.94e-04 13s
74 1.23950929e+00 -1.58639935e+01 6.56e-06 4.37e-08 1.70e-04 13s
75 1.00016329e+00 -1.12424539e+01 3.43e-06 2.28e-08 1.22e-04 14s
76 7.94250834e-01 -6.83743613e+00 1.30e-06 8.68e-09 7.57e-05 14s
77 6.07205738e-01 -2.83440051e+00 2.09e-07 1.39e-09 3.41e-05 14s
78 6.03943103e-01 -2.75740497e+00 1.89e-07 1.26e-09 3.33e-05 14s
79 5.56924660e-01 -9.73106797e-01 6.70e-09 4.46e-11 1.52e-05 14s
80 5.17702916e-01 -6.88815374e-02 4.10e-10 2.67e-12 5.81e-06 14s
81 5.05706078e-01 4.30962307e-01 1.94e-11 1.98e-13 7.41e-07 15s
82 5.04461712e-01 4.86505491e-01 2.12e-11 4.20e-14 1.78e-07 15s
83 5.04083134e-01 5.03659707e-01 9.16e-12 7.91e-16 4.20e-09 15s
84 5.04074193e-01 5.04073703e-01 2.08e-11 1.58e-15 4.85e-12 15s

Barrier solved model in 84 iterations and 15.15 seconds
Optimal objective 5.04074193e-01


User-callback calls 36246, time in user-callback 0.07 sec

However, when I am at iteration 24, It is:

Gurobi Optimizer version 9.1.1 build v9.1.1rc0 (mac64)
Thread count: 8 physical cores, 16 logical processors, using up to 16 threads
Optimize a model with 164810 rows, 107382 columns and 1376563 nonzeros
Model fingerprint: 0xd65e127c
Model has 1876 quadratic objective terms
Coefficient statistics:
Matrix range [3e-08, 9e+04]
Objective range [4e-11, 2e+01]
QObjective range [7e-02, 2e+00]
Bounds range [6e-01, 3e+05]
RHS range [1e-01, 2e+05]
Warning: Model contains large matrix coefficient range
Consider reformulating model or setting NumericFocus parameter
to avoid numerical issues.
Presolve removed 109633 rows and 60916 columns
Presolve time: 0.46s
Presolved: 55177 rows, 46466 columns, 818354 nonzeros
Presolved model has 1876 quadratic objective terms
Ordering time: 0.65s
Elapsed ordering time = 5s
Elapsed ordering time = 6s
Elapsed ordering time = 8s
Elapsed ordering time = 10s
Elapsed ordering time = 10s
Ordering time: 11.01s

Barrier statistics:
AA' NZ : 3.933e+07
Factor NZ : 7.830e+08 (roughly 6.0 GBytes of memory)
Factor Ops : 1.651e+13 (roughly 70 seconds per iteration)
Threads : 8

Objective Residual
Iter Primal Dual Primal Dual Compl Time
0 6.63854293e+13 -1.94541185e+14 3.55e+07 2.42e+05 7.75e+10 120s
1 1.44911927e+13 -1.37385643e+14 1.50e+07 1.02e+05 3.47e+10 228s

As you can see, I stop the solver because it is clearly something different happened at iteration 24 and makes QP behaves badly.

Could anyone tell me how can I fix this?

Many thanks!

Hongyu
  • 183
  • 5
  • 1) According to the solver it's not the same QP problem. So start looking into that! 2) There are already warnings which of course can sometimes become a problem and sometimes not - **edit**: Internally, the reason is the sparsity-structure / factorization within the IPM. The difference between those iterations (it's not the same model!) has a very bad impact there and each iteration will be much slower. Probably some *dense columns or rows*. (see Barier statistics) – sascha Jan 30 '21 at 16:03
  • Thanks @sascha for your prompt reply! It is my first time asking question here. Sorry I didn't make the question clear. It is indeed not the same mode! I added one extra constrain (cut) in the model at every iteration. What I try to say is the same structure. But I could not find why after doing this 23 iterations, it becomes extreme slow. Regarding dense columns, I use CPLEX to solve the same problem and there are 61 dense columns. I searched online and know dense columns affect algorithm seriously. Is there anyway to reduce it? I don't think I can reformulate the model... – Hongyu Jan 30 '21 at 16:22
  • The logs show gurobi, now you talk about cplex. I'm afraid that there is not much to do here without knowing what exactly is being solved and happening. The logs say a lot though: 756 constraints are added between those two iterations wich is not equal to 1 (maybe your modelling framework hides some steps?). Those added constraints are quite dense as 37810 non-zeros are added. Maybe an expert can read something out of this (e.g. AAT is slightly worse but the factorization is totally collapsing). Consider gur/cplex-support or try to change factorization options. Maybe you need to fix your model – sascha Jan 30 '21 at 16:30
  • This is how the optimization world looks like! You could try to reformulate your model to a linear one and than use the linear solution as a starting point for QP, perhaps with some `TimeLimit` setting. For such complex models performance will also vary greatly across solvers. – Przemyslaw Szufel Jan 30 '21 at 16:42
  • @PrzemyslawSzufel Thanks for the suggestion! – Hongyu Jan 30 '21 at 21:53

0 Answers0