Duplicate constraints in Pyomo model

Pyomo [1]  is a popular Python based modeling tool. In [2] a question is posed about a situation where a certain constraint takes more than 8 hours to generate. As we shall see, the reason is that extra indices are used.

A simple example

The constraint \[y_i = \sum_j x_{i,j} \>\>\>\forall i,j\] is really malformed. The extra \(\forall j\) is problematic. What does this mean? One could say, this is wrong. We can also interpret this differently. Assume the inner \(j\) is scoped (i.e. local). Then we could read this as: repeat the constraint \(y_i = \sum_j x_{i,j}\), \(n\) times. Here \(n=|J|\) is the cardinality of set \(J\).

The GAMS fragment corresponding to this example, shows GAMS will object to this construct:

11  equation e(i,j);   12  e(i,j)..  y(i) =e= sum(j, x(i,j)); ****                          $125 **** 125  Set is under control already   13   **** 1 ERROR(S)   0 WARNING(S)

The Pyomo equivalent can look like:

def eqRule(m,i,j):     return m.Y[i] == sum(m.X[i,j] for j in m.J); model.Eq = Constraint(model.I,model.J,rule=eqRule)

This fragment is a bit more difficult to read, largely due to syntactic clutter. But in any case: Python and Pyomo accepts this constraint as written. To see what is generated, we can use

model.Eq.pprint()

This will show something like:

Eq : Size=6, Index=Eq_index, Active=True     Key          : Lower : Body                                     : Upper : Active     ('i1', 'j1') :   0.0 : Y[i1] - (X[i1,j1] + X[i1,j2] + X[i1,j3]) :   0.0 :   True     ('i1', 'j2') :   0.0 : Y[i1] - (X[i1,j1] + X[i1,j2] + X[i1,j3]) :   0.0 :   True     ('i1', 'j3') :   0.0 : Y[i1] - (X[i1,j1] + X[i1,j2] + X[i1,j3]) :   0.0 :   True     ('i2', 'j1') :   0.0 : Y[i2] - (X[i2,j1] + X[i2,j2] + X[i2,j3]) :   0.0 :   True     ('i2', 'j2') :   0.0 : Y[i2] - (X[i2,j1] + X[i2,j2] + X[i2,j3]) :   0.0 :   True     ('i2', 'j3') :   0.0 : Y[i2] - (X[i2,j1] + X[i2,j2] + X[i2,j3]) :   0.0 :   True

We see for each \(i\) we have three duplicates. The way to fix this is to remove the function argument \(j\) from eqRule:

def eqRule(m,i):     return m.Y[i] == sum(m.X[i,j] for j in m.J); model.Eq = Constraint(model.I,rule=eqRule)

After this, model.Eq.pprint() produces

Eq : Size=2, Index=I, Active=True     Key : Lower : Body                                     : Upper : Active      i1 :   0.0 : Y[i1] - (X[i1,j3] + X[i1,j2] + X[i1,j1]) :   0.0 :   True      i2 :   0.0 : Y[i2] - (X[i2,j3] + X[i2,j2] + X[i2,j1]) :   0.0 :   True

This looks much better.

The original problem

The constraint in the original question was:

def period_capacity_dept(m, e, j, t, dp):     return sum(a[e, j, dp, t]*m.y[e,j,t] for (e,j) in model.EJ)<= K[dp,t] + m.R[t,dp] model.period_capacity_dept = Constraint(E, J, T, DP, rule=period_capacity_dept)

Using the knowledge of the previous paragraph we know this should really be:

def period_capacity_dept(m, t, dp):     return sum(a[e, j, dp, t]*m.y[e,j,t] for (e,j) in model.EJ)<= K[dp,t] + m.R[t,dp] model.period_capacity_dept = Constraint(T, DP, rule=period_capacity_dept)

Pyomo mixes mathematical notation with programming. I think that is one of the reasons this bug is more difficult to see. In normal programming, adding an argument to a function has an obvious meaning. However in this case, adding e,j means in effect: \(\forall e,j\). If \(e\) and \(j\) belong to large sets, we can easily create a large number of duplicates.

References