For a Bi-level Mixed Integer Linear Program with integer variables in the lower, can I use KKT conditions to reduce the problem to a single level?

Question

For example, my optimization formulation looks something like this:

max $-10y-x$

s.t. $y=$ arg {min $20y-25x\leq 30;2y+x\leq 10;-y+2x\leq 15;10y+2x\geq15$}

$y$ integer

In order to convert this to a single level problem, can I formulate the Lagrangian and use KKT? If not, what are some other approaches to covert BMILP into single level problems?

Answer 1

No, it is a much harder problem.

There are some good presentations here https://coral.ie.lehigh.edu/~ted/research/presentations/