Continuous variable behaviour on MIP
AnsweredHello,
I have a MIP model with a vector of integer variables and a single continuous one that appears on the objective function, what is the expected behaviour for the continuous variable when a heuristic solution is found? It's the optimal value (of the continuous variable) for the heuristic integer variables?
For context, I have a callback where I hash the pair {integer vector: continuous variable}, and I found that a specific vector was visited twice, the first time as a heuristic solution with a given continuous value, and after some iterations it was visited again after branching but now with a better value for the continuous variable.

Hi Steffan,
Heuristics provide feasible solution for MIPs. This means that the pair of integers and the single continuous variable your mentioned, determine one feasible point for your problem. It is not guaranteed to be the global optimum for your problem unless the lower bound converged with a MIPGap of 0.
It is very well possible that one heuristic finds a solution with integer vector \(I\) and continuous variable \(c_1\) while a different heuristic finds a feasible point a bit later with better objective value given as the same integer vector \(I\) and a different continuous variable \(c_2 \neq c_1\). This may happen often as some heuristics use available feasible points to generate new ones.
Best regards,
Jaromił1 
Thanks Jaromił!
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