Gurobi Warning and Inconsistency in Optimal Value [some integer variables take values larger than the maximum supported value (2000000000)] - gurobi

Question

I am using Gurobi version 8.1.0 and Python API version 3.6 to solve MIP problems. I have two models in which I believe that their global optima are equal. However, I found out that they are not equal in one of my simulations. I then tried to warm-start the model that I believe the solution is incorrect (model-1) with the solution from another model (model-2). In other words, the problem is to maximize the objective function and the objective value of model-1 is 42.3333, but I believe it should be 42.8333. Therefore, I use the solution from model-2 with the objective value of 42.8333 to warm-start to model-1.
What is weird is that the solution from model-2 should not be feasible to model-1 since the objective value is greater than 42.3333 and the problem is maximization. However, it turns out that it is a feasible warm start and now the optimal value of model-1 is 42.8333. How can the same model have multiple optima?
Changed value of parameter timeLimit to 10800.0
Prev: 1e+100 Min: 0.0 Max: 1e+100 Default: 1e+100
Changed value of parameter LogFile to output/inconsistent_Model-1.log
Prev: gurobi.log Default:
Optimize a model with 11277 rows, 15150 columns and 165637 nonzeros
Model has 5050 general constraints
Variable types: 0 continuous, 15150 integer (5050 binary)
Coefficient statistics:
Matrix range [1e+00, 1e+00]
Objective range [1e-02, 1e+00]
Bounds range [1e+00, 1e+00]
RHS range [1e+00, 5e+01]
Presolve removed 6167 rows and 7008 columns
Presolve time: 0.95s
Presolved: 5110 rows, 8142 columns, 37608 nonzeros
Presolved model has 3058 SOS constraint(s)
Variable types: 0 continuous, 8142 integer (4403 binary)
Warning: Markowitz tolerance tightened to 0.0625
Warning: Markowitz tolerance tightened to 0.125
Warning: Markowitz tolerance tightened to 0.25
Warning: Markowitz tolerance tightened to 0.5
Root relaxation: objective 4.333333e+01, 4856 iterations, 2.15 seconds
Nodes | Current Node | Objective Bounds | Work
Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time
0 0 43.33333 0 587 - 43.33333 - - 3s
0 0 43.26667 0 243 - 43.26667 - - 4s
0 0 43.20000 0 1282 - 43.20000 - - 4s
0 0 43.20000 0 567 - 43.20000 - - 4s
0 0 43.18333 0 1114 - 43.18333 - - 5s
0 0 43.16543 0 2419 - 43.16543 - - 5s
0 0 43.15556 0 1575 - 43.15556 - - 5s
0 0 43.15333 0 2271 - 43.15333 - - 5s
0 0 43.13333 0 727 - 43.13333 - - 5s
0 0 43.12778 0 1698 - 43.12778 - - 5s
0 0 43.12500 0 1146 - 43.12500 - - 5s
0 0 43.12500 0 1911 - 43.12500 - - 6s
0 0 43.11927 0 1859 - 43.11927 - - 6s
0 0 43.11845 0 2609 - 43.11845 - - 7s
0 0 43.11845 0 2631 - 43.11845 - - 7s
0 0 43.11845 0 2642 - 43.11845 - - 7s
0 0 43.11845 0 2462 - 43.11845 - - 8s
0 0 43.11845 0 2529 - 43.11845 - - 8s
0 0 43.11845 0 2529 - 43.11845 - - 9s
0 2 43.11845 0 2531 - 43.11845 - - 14s
41 35 43.09874 17 957 - 43.09874 - 29.4 15s
94 84 42.93207 33 716 - 43.09874 - 22.1 31s
117 101 42.91940 40 2568 - 43.09874 - 213 37s
264 175 infeasible 92 - 43.09874 - 133 73s
273 181 infeasible 97 - 43.09874 - 277 77s
293 191 42.42424 17 1828 - 43.09874 - 280 90s
369 249 42.40111 52 2633 - 43.09874 - 311 105s
383 257 42.39608 59 3062 - 43.09874 - 329 152s
408 265 42.39259 65 2819 - 43.09874 - 386 162s
419 274 41.51399 66 2989 - 43.09874 - 401 170s
454 282 41.29938 71 3000 - 43.09874 - 390 182s
462 280 infeasible 74 - 43.09874 - 423 192s
479 287 infeasible 78 - 43.09874 - 419 204s
498 293 40.51287 81 2564 - 43.09874 - 435 207s
526 307 40.16638 86 2619 - 43.09874 - 419 227s
584 330 42.63100 33 621 - 43.09874 - 404 236s
628 333 infeasible 37 - 43.09874 - 394 252s
661 345 42.37500 26 25 - 43.09874 - 396 288s
684 353 infeasible 30 - 43.09874 - 426 290s
842 370 infeasible 69 - 43.09874 - 348 306s
944 379 infeasible 86 - 43.09874 - 321 370s
1009 395 42.36667 22 25 - 43.09874 - 350 409s
* 1031 243 3 42.3333333 43.09874 1.81% 343 409s
1056 203 43.00000 19 141 42.33333 43.09874 1.81% 362 411s
1194 222 cutoff 23 42.33333 43.00000 1.57% 325 430s
1199 219 cutoff 25 42.33333 43.00000 1.57% 349 450s
1202 212 cutoff 29 42.33333 43.00000 1.57% 361 472s
1211 200 infeasible 47 42.33333 42.91851 1.38% 380 498s
1226 169 infeasible 43 42.33333 42.91471 1.37% 395 511s
Cutting planes:
Gomory: 2
Cover: 15
Implied bound: 1
Clique: 26
MIR: 17
Inf proof: 1
Zero half: 8
Explored 1426 nodes (502432 simplex iterations) in 512.68 seconds
Thread count was 4 (of 4 available processors)
Solution count 1: 42.3333
Optimal solution found (tolerance 1.00e-04)
Warning: some integer variables take values larger than the maximum
supported value (2000000000)
Best objective 4.233333333333e+01, best bound 4.233333333333e+01, gap 0.0000%
In addition to the above, I also received this warning:
"Optimal solution found (tolerance 1.00e-04)
Warning: some integer variables take values larger than the maximum
supported value (2000000000)". What does it mean? Thank you so much!

Answer 1

It looks like you are encountering some numerical troubles. The root relaxation required an increased Markowitz tolerance, which indicates an ill-conditioned matrix. This may lead to inconsistencies as you have observed in the two different "optimal" solutions.
The warning about too large values means that there are integer variables with solution values so large that the integer feasibility tolerance can not reliably be checked anymore. If you have a variable with solution value in the range of 1e+9 it probably doesn't matter anymore whether they are integer or not. So you could probably also simplify your model by making them continuous variables.
You should check for violations in the two solutions for both models (see here) to see how feasible the solutions actually are.