Different best objective reports for mixed ILP

[rootjalex]

For the following example, Gurobi is reporting a best objective value of 4:

using JuMP
using Gurobi
model = Model(Gurobi.Optimizer);

@variable(model, extent)
@variable(model, y)
@variable(model, yo)
@variable(model, _t0)
@variable(model, _t4)
@variable(model, _t2, Bin)
@variable(model, _t3)

@constraint(model, _t0 <= ((((y * 4) + yo) * 2) + 2))
@constraint(model, _t0 <= extent)
@constraint(model, _t4 <= (((y * 4) + yo) * 2))
@constraint(model, _t4 <= extent)

@constraint(model, _t2 => { (1 <= yo) })
@constraint(model, !_t2 => { (yo <= 0) })
@constraint(model, _t2 => { _t3 == (_t4 + 4)})
@constraint(model, !_t2 => { _t3 == (((y * 4) + yo) * 2)})

@objective(model, Max, ((_t0 - _t3) + 4))
print(model)
optimize!(model)

Output:

Gurobi Optimizer version 9.1.1 build v9.1.1rc0 (mac64)
Thread count: 6 physical cores, 12 logical processors, using up to 12 threads
Optimize a model with 4 rows, 7 columns and 10 nonzeros
Model fingerprint: 0x2ca96aa2
Model has 4 general constraints
Variable types: 6 continuous, 1 integer (1 binary)
Coefficient statistics:
  Matrix range     [1e+00, 8e+00]
  Objective range  [1e+00, 1e+00]
  Bounds range     [0e+00, 0e+00]
  RHS range        [2e+00, 2e+00]
Found heuristic solution: objective 4.0000000
Presolve removed 4 rows and 6 columns
Presolve time: 0.00s

Explored 0 nodes (0 simplex iterations) in 0.00 seconds
Thread count was 1 (of 12 available processors)

Solution count 1: 4 
No other solutions better than 0

Model is unbounded
Best objective 4.000000000000e+00, best bound -, gap -

User-callback calls 45, time in user-callback 0.00 sec

Now, if I add a single constraint to force a higher best objective:

@constraint(model, extent == 2)

Then a higher best objective is obtained:

Gurobi Optimizer version 9.1.1 build v9.1.1rc0 (mac64)
Thread count: 6 physical cores, 12 logical processors, using up to 12 threads
Optimize a model with 5 rows, 7 columns and 11 nonzeros
Model fingerprint: 0xfdbf4857
Model has 4 general constraints
Variable types: 6 continuous, 1 integer (1 binary)
Coefficient statistics:
  Matrix range     [1e+00, 8e+00]
  Objective range  [1e+00, 1e+00]
  Bounds range     [0e+00, 0e+00]
  RHS range        [2e+00, 2e+00]
Presolve added 2 rows and 4 columns
Presolve time: 0.00s
Presolved: 7 rows, 11 columns, 19 nonzeros
Presolved model has 4 SOS constraint(s)
Variable types: 8 continuous, 3 integer (3 binary)
Found heuristic solution: objective 6.0000000

Root relaxation: unbounded, 2 iterations, 0.00 seconds

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

     0     0  postponed    0         6.00000          -      -     -    0s
     0     0  postponed    0         6.00000          -      -     -    0s
     0     2  postponed    0         6.00000          -      -     -    0s

Explored 3 nodes (9 simplex iterations) in 0.00 seconds
Thread count was 12 (of 12 available processors)

Solution count 1: 6 

Model is unbounded
Best objective 6.000000000000e+00, best bound 6.000000000000e+00, gap 0.0000%

User-callback calls 60, time in user-callback 0.00 sec

I am a bit confused as to why the first example is reporting a best objective for an unbounded model in the first place, instead of simply not calculating/reporting an objective value.

[odow]

Gurobi is reporting a best objective value of 4:

This is not what it is saying. Before proving the model is unbounded, Gurobi detected an objective of 4. This is expected behavior and not a bug.

Use termination_status(model) and primal_status(model) to understand the solution returned by the solver. In this case, you get:

julia> termination_status(model)
DUAL_INFEASIBLE::TerminationStatusCode = 3

julia> primal_status(model)
NO_SOLUTION::ResultStatusCode = 0

The fact that

julia> objective_value(model)
4.0

is meaningless because the problem is unbounded.

[rootjalex]

Ah, that makes sense, thank you!