Binary variables non exactly binary in Python

[Nicola Rosada]

Hi,

I'm using some binary variables in Gurobi for Python:

self.y = {}

for m in range(1,self.NUM_OF_TM+1):

    for n in range(1,self.N+1):

        self.y[m,n] = self.mod.addVar(name='y_%s_%s' % (m,n), vtype=GRB.BINARY)

after have solved the model I get the variables with getAttr:

self.y_solution = self.mod.getAttr('x', self.y)

print 'self.y_solution:'

print self.y_solution

I obtain a dictionary with float variables and in most of the cases the values are 0.0 or 1.0 as expected. But I found very few instances where the values are not exactly 0 or 1, but they are something like 1e-8 or 0.99999999.

It seems strange, is there something wrong in what I'm doing?

Thank you all!

Bye

[Tobias Achterberg]

This can happen due to feasibility and integrality tolerances. You will also
find that solution that Gurobi (as all floating-point based MIP solvers)
provides may slightly violate your constraints.

The reason is that floating-point numeric as implemented in the CPU hardware is
not exact. Rounding errors can (and usually will) happen. As a consequence, MIP
solvers use tolerances within which a solution is still considered to be
correct. The default tolerance for integrality in Gurobi is 1e-5, the default
feasibility tolerance is 1e-6. This means that Gurobi is allowed to consider a
value that is at most 1e-5 away from an integer to still be integral, and it is
allowed to consider a constraint that is violated by at most 1e-6 to still be
satisfied.

You can tighten the tolerances up to 1e-9, but this does not get rid of the
fundamental issue that the CPU makes errors when calculating with floating point
numbers.


Best regards,

Tobias

[Nicola Rosada]

I knew about the tolerance for the floating point numbers, but I thought that this problem didn't happen for binary and integer values that can be correctly stored in memory.

Thank you very much for the answer.

Best regards,

Nicola