I am curious about the above-mentioned relationship. Say you have a bilevel optimization problem of the type Stackelberg; one approach is to transform the problem into a single-level optimization problem with the lower-level problem formulated as a first-order optimality condition. Can the latter only be done if the optimization problem is linear? Is this type of formulation an MPEC?
Thanks in advance!
You need to be able to express the solution to the inner level optimization problem as the solution to the Karush Kuhn Tucker (KKT) first order optimality conditions. This can be done for any convex optimization problem satisfying a KKT constraint qualification, in which case 1st order KKT conditions are necessary and sufficient for global optimality of the inner problem.
The complementarity constraint in the KKT conditions is what makes a problem an MPEC. The outer level optimization problem can be anything, with "any" constraints, to which the 1st order KKT conditions for the inner problem are added. Although the difficulty of (numerically) solving the resulting MPEC depends on what the outer level problem is, as well as the KKT conditions for the inner level problem.