How to add a binary variable without objective value using the Java API

Dear all,

my question is how to create binary variable in Gurobi without assigning a cost/objective value to it. I use the Java API method `GRBModel.addVar(...)` to add variables to my model; this is the function signature([REF](https://www.gurobi.com/documentation/9.5/refman/java_model_addvar.html#javamethod:GRBModel.addVar)):

GRBVar addVar (double lb, double ub, double obj, char type, String name)

This method only alows adding variables with associated objective value (i.e. the `obj` argument). 

However, for the problem I am trying to solve, I need to introduce an auxiliary binary variable (see below for details), which I use to "deactivate" a constraint (lets call this variable \(d\)). I do not want the model to incur a cost, when the auxiliary variable \(d\) is selected.

Is this possible and I do this?

Here is some more details of my particular optimization problem:

I have a set of "main" constraints like this one. Let's consider a single instance for simplicity:

\(\sum \beta_j \cdot b_j  \leq \sum_i \alpha_i \cdot a_i + M \cdot d\)

where \(a_i\), \(b_j\) are binary variables with respective objective values \(\alpha_i\), \(\beta_j\). \(d\) is the auxiliary binary variable mentioned earlier.

I want this constraint become "inactive" when \(\sum_i a_i = 0\). To do this I add this auxiliary constraint:

\(\sum_i  a_i + d = 1\)

Thus \(d=1\) when all \(a_i=0\) and vice-versa and hence the constraint above becomes:

\(\sum \beta_j \cdot b_j  \leq M \cdot d\)

where in my code I set \(M\) to the maximum value allowed by double values in Java (i.e. `M=Double.MAX_VALUE`). Thus the sum \(\sum \beta_j \cdot b_j\) is unconstraint, when \(\sum_i a_i = 0\).

But (as said above) I do not want the selection of \(d\) be penalized or favored by an objective value. It should be just auxiliary variable/dummy variable without associated objective value ...

Thank you in advance,

Michael

PS: I found [this example](https://support.gurobi.com/hc/en-us/community/posts/360057913451/comments/360009559012) on the Gurobi forum, where apparantly a binary variable is created without associated cost using the Python API.

PPS: How do I format Math equations on this forum?