CPLEX not substituting equality correctly?

Question

I am quite new to CPLEX and I am writing a very simple model that CPLEX does not want to satisfy. I know my model is "verbose" as I have variables that simply equal other variables, but it is my first step to a more complicated model so I want it this way. I don't see why would this upset CPLEX.

The model I have is:

Minimize
obj: v1
Subject To

l1: i_AB1 + i_AC1 - i_AB2 - i_AC3 = 1
l2: - i_AB1 + I_BB = 0
l3: I_CA + I_CC = 0


e5:   i_AB2 + i_BD2 - I_BB = 0

e8:   - i_AC1 - I_CA = 0
e9:   i_AC3 + i_CD3 - I_CC = 0


\Indicator constraints

\For each connection XY

c1:  bAB = 1-> i_AB1 - 1 v1 = 0
c2:  bAB = 1-> i_AB2 - 1 v2 = 0

c5:  bAC = 1-> i_AC1 - 1 v1 = 0
c6:  bAC = 1-> i_AC3 - 1 v3 = 0

c9:  bBD = 1-> i_BD2 - 1 v2 = 0

c13: bCD = 1-> i_CD3 - 1 v3 = 0

c15: bAB = 1
c16: bAC = 1
c17: bBD = 1
c18: bCD = 1


Bounds
    0 <= v1 <= 1000
-1000 <= v2 <= 1000
-1000 <= v3 <= 1000


General

Binaries
bAB bAC bBD bCD
End

This, apparently does not have a solution (it does, or at least that is my intention, but CPLEX says no!).

But then I substitute the equation e8 into l3 and I get the solution I wanted! Here is the code:

Minimize
obj: v1
Subject To


\budget: 

l1: i_AB1 + i_AC1 - i_AB2 - i_AC3 = 1
l2: - i_AB1 + I_BB = 0
l3: - i_AC1  + I_CC = 0

e5:   i_AB2 + i_BD2 - I_BB = 0

\Row C
\e8:   - i_AC1 - I_CA = 0
e9:   i_AC3 + i_CD3 - I_CC = 0


\Indicator constraints

\For each connection XY

c1:  bAB = 1-> i_AB1 - 1 v1 = 0
c2:  bAB = 1-> i_AB2 - 1 v2 = 0

c5:  bAC = 1-> i_AC1 - 1 v1 = 0
c6:  bAC = 1-> i_AC3 - 1 v3 = 0

c9:  bBD = 1-> i_BD2 - 1 v2 = 0

c13: bCD = 1-> i_CD3 - 1 v3 = 0

c15: bAB = 1
c16: bAC = 1
c17: bBD = 1
c18: bCD = 1


Bounds
    0 <= v1 <= 1000
-1000 <= v2 <= 1000
-1000 <= v3 <= 1000


General

Binaries
bAB bAC bBD bCD
End

Both are, to my eyes, the exact same model. What am I doing wrong so that the first model does not have a solution even though it seems equivalent to the second which does have a solution?

Btw, the solution is:

Populate: phase I 
Tried aggregator 2 times.
MIP Presolve eliminated 4 rows and 4 columns.
Aggregator did 11 substitutions.
All rows and columns eliminated.
Presolve time =    0.00 sec.

Populate: phase II 
Solution status: 129.
Objective value of the incumbent: 1
Incumbent: Column v1:  Value =                 1
Incumbent: Column i_AB1:  Value =                 1
Incumbent: Column i_AC1:  Value =                 1
Incumbent: Column i_AB2:  Value =               0.5
Incumbent: Column i_AC3:  Value =               0.5
Incumbent: Column I_BB:  Value =                 1
Incumbent: Column I_CC:  Value =                 1
Incumbent: Column i_BD2:  Value =               0.5
Incumbent: Column i_CD3:  Value =               0.5
Incumbent: Column bAB:  Value =                 1
Incumbent: Column v2:  Value =               0.5
Incumbent: Column bAC:  Value =                 1
Incumbent: Column v3:  Value =               0.5
Incumbent: Column bBD:  Value =                 1
Incumbent: Column bCD:  Value =                 1

The solution pool contains 1 solutions.
0 solutions were removed due to the solution pool relative gap parameter.
In total, 1 solutions were generated.
The average objective value of the solutions is 1.

Solution        Objective   Number of variables
                value       that differ compared to
                            the incumbent
p1              1         0 / 15

The problem itself is not even a MIP (because I fixed my booleans in this initial version, but it will be a proper MIP). Does this change something? I really don't see what the issue is.

Thanks

Answer 1

I believe the problem is due to the fact that by default I_CA has a lower bound of 0. In the Bounds section in the documentation for the LP format it says

Upper and lower bounds may also be entered separately ... with the default lower bound of 0 (zero) and the default upper bound of +∞ remaining in effect until the bound is explicitly changed.

If you add I_CA free to the bounds section for your first LP file, then it becomes feasible.