Boolean comparison of decision variable with parameter
Hello, I want to create a sort of indicator function to use in my objective function. Here is the set up:
I have x_i for i from 1 to 10 as my decision variables. I have a list of lists (let's call it y, with one of these lists being called y_r) as an input; for example, 100 lists of 10 numbers, which are all randomly generated decision variables for which I have to compare my solution. If x_i is larger than y_r_i, I add a certain value v_i to my objective value. Else, I add 0.
This is my current code (r is given in this function):
gb.quicksum(v[i]*x[i] if x[i] > y[r][i] else 0 for i in range(10))
The error I get is:
TypeError: '>' not supported between instances of 'Var' and 'int'
i.e. due to x[i] being a decision variable in my Gurobi model and y[r][i] being an input integer.
How can I make this objection function work?
Kind regards,
Mike
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Hello Mike,
The modelling approach will depend on whether you want to minimize or maximize your objective.
For example, if you want to maximize, with r fixed, and v's being non-negative, to build something like
> gb.quicksum(v[i]*x[i] if x[i] >= y[r][i] else 0 for i in range(10))
or essentially, for simplicity dropping sub-index r, add terms of the form
v[i]*x[i] if x[i] >= y[i]
one can use the so-called semi-continuous variables construct, see here https://www.gurobi.com/documentation/9.0/refman/variables.html
Then you can introduce an auxiliary semi-continuous z[i] with lower bound y[i] and upper bound GRB.INFINITY, unconditionally add the term
(*) v[i] * z[i]
to the objective, and add a constraint
(**) x[i] >= z[i]
Note if x[i] ends up smaller than y[i], due to (**) and z[i] being semi-continuous, z will take value 0 making no contribution to the objective. On the other hand, if x[i] ends up greater or equal than y[i], due to maximizing the objective and (**) you will essentially be adding v[i]*x[i] to the objective function.
Hope this helps.
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