Solving MILP lower-level and non-convex MIQCQP upper-level with BilevelJuMP

[Paulnkk]

Hey,

I wanted to as if it is possible to solve the problem class mentioned above with BilevelJuMP?

Thank you,

Paul

[Paulnkk]

Edit: Alternatively, I can model the lower level as non-convex quadratic (I have one coupling constraint of the form x*y = 0 in the lower-level with x and y as variables of the lower-level). The upper-level is the same as above

[joaquimg]

Hi!
Nowadays you have 2 general options:
1 - upper level LP, QP, MIQP, MIP or NLP. Lower level LP or Conic QP.
2 - upper level LP or MIP. Lower level LP or MIP. This second way requires MiBS

[Paulnkk]

@joaquimg, thanks a lot for your response! I think I will not be able to solve my bi-level problem with currently available bi-level optimization software since I assume you refer to MIP as MILP and as mentioned above my upper-level problem is non-convex, mixed-integer and MIQCQP. From the MiBS docu it seems that the optimizer can just solve for upper and lower level MILP

[joaquimg]

Yes, I meant MILP with MIP.

MILP lower levels are extremely hard.

In theory,
You can approximate an NLP with piecewise linear things in a MILP, but that might lead to an extremely hard MILP.

[Paulnkk]

@joaquimg Thanks a lot for your help so far, I have one additional question: If I have lower-level and upper-level MILP, I can use the software to calculate a solution, right ? Do you know any literature concerning optimality conditions for this situation I can use in my paper ?

Edit1: I have seen some literature and it seems that in the upper-and lower-level formulations, the decision variables should be separated. Is that a necessary assumption ?

Edit2: My MILPs would also just be MILPs in their decision variables

[joaquimg]

You can start from the MIBS paper: https://link.springer.com/article/10.1007/s12532-020-00183-6