问题
I'm using Julia's wonderful JuMP package to solve a linear program with Gurobi 6.0.4 as a solver. The objective function is a sum of decision variables, clearly defined as nonnegative, and the problem requires it to be minimized. For some reason, Gurobi thinks the model is unbounded.
Here is the definition of the variables and the objective:
@defVar(model, delta2[i=irange,j=pair[i]] >= 0)
@setObjective(model, Min, sum{delta2[i,j], i=irange, j=pair[i]})
Strange observation #1: although this is a minimization problem, the log for Gurobi's BarrierSolve method clearly shows the objective function increasing at each iteration. In addition, Gurobi seems to pull a switcheroo between the number of rows and the number of columns. Before the presolve step, the model has 50k rows and 25k columns. After the presolve step (which removes less than 1k rows and columns), we have 24k rows and 50k columns. The log looks like this:
Optimize a model with 50422 rows, 24356 columns and 1846314 nonzeros
Coefficient statistics:
Matrix range [1e-04, 2e+00]
Objective range [1e+00, 1e+00]
Bounds range [9e-02, 2e+02]
RHS range [6e+00, 4e+03]
Presolve removed 164 rows and 635 columns
Presolve time: 0.79s
Presolved: 24192 rows, 49787 columns, 1836247 nonzeros
Ordering time: 1.60s
Strange observation #2: BarrierSolve eventually terminates with the status InfeasibleOrUnbounded. It then suggests setting InfUnbdInfo=1 and using Gurobi's homogeneous BarrierSolve method by setting BarHomogeneous=1. When I do both of these things, the objective function keeps increasing(!) and the barrier log looks like this:
Objective Residual
Iter Primal Dual Primal Dual Compl Time
0 -6.95693531e+06 1.94975493e+02 1.10e+01 9.79e+02 1.39e+03 4s
1 -3.18487510e+06 7.02065119e+06 5.50e-01 5.57e+02 3.45e+02 5s
2 -8.43175324e+05 2.31465924e+06 4.81e-02 1.60e+02 9.32e+01 6s
3 -2.37967254e+05 6.66124613e+05 6.51e-03 3.69e+01 2.35e+01 8s
4 -7.49693243e+04 1.81252940e+05 1.64e-03 9.49e+00 6.46e+00 9s
5 -3.20211009e+04 8.98339452e+04 6.25e-04 5.30e+00 3.11e+00 10s
6 -1.04312874e+04 5.17677474e+04 2.06e-04 3.06e+00 1.65e+00 11s
7 4.58252702e+02 4.04538611e+04 1.24e-04 2.19e+00 1.23e+00 12s
8 3.40831629e+04 5.42543944e+04 7.65e-05 1.87e+00 1.54e+00 13s
9 3.10110459e+05 2.25902448e+05 5.50e-05 1.87e+00 6.81e+00 15s
10 1.59299448e+06 9.88980682e+05 5.73e-05 1.85e+00 3.37e+01 16s
11* 1.88981433e+07 1.28711401e+07 2.93e-06 6.92e-01 1.14e-03 17s
12* 1.65096505e+08 3.73470456e+08 5.57e-06 5.73e-02 1.40e-04 18s
13* 7.18252597e+09 3.21890978e+09 2.62e-06 2.01e-03 4.60e-06 20s
14* 1.15822505e+12 7.53575462e+10 1.50e-05 6.18e-06 8.50e-09 21s
15* 1.08512896e+13 2.57735417e+12 2.92e-06 6.99e-08 1.22e-10 22s
16* 3.03152292e+14 7.54681485e+13 1.21e-07 7.50e-10 1.28e-12 23s
Barrier performed 16 iterations in 23.41 seconds
Unbounded model
I don't understand how a linear program can be unbounded when it involves the minimization of a sum of nonnegative variables. Is this an issue with Gurobi or did I do something wrong in setting up the LP ? I suspect it might be some sort of numerical error, but I'm not sure how to troubleshoot it.
EDIT: I found a partial fix to the problem by loosening some constraints and making the feasibility region artificially better. It seems the problem was really a feasibility issue and not an unboundedness issue, which means Gurobi might actually have been referring to an unboundedness of the dual ?
Thanks for your help!
回答1:
Your problem is poorly conditioned due to "narrow" range constraints. If they really need to be range constraints, consider rescaling your problem. If they are more like "approximate" equality constraints, consider adding a slack variable and making it an actual equality constraint, and put a penalty on the slacks in the objective.
来源：https://stackoverflow.com/questions/31542234/gurobi-reports-unbounded-model-despite-mathematical-impossibility