As it can be observed, the lower-le

As it can be observed, the lower-level problem is a parametric optimization problem which is set by the upper-level decision maker. The latter has to determine the lower-level rational set, i.e., Ψ(x) to evaluate its own objective function. Ψ(x) is a set since the lower-level decision maker may have several rational reactions for the same lower-level instance. These reactions are all optimal for the lower-level decision maker which is not the case for the upper-level decision maker. If for a given upper-level decision x, the lower-level decision maker has many optimal choices, we need to turn to set-valued optimization and more precisely to point to set mapping. Another solution is to consider two special cases：