A note on the definition of a linear bilevel programming solution

An alternative definition of the linear bilevel programming problem BLP has recently been proposed by Lu, Shi, and Zhang. This note shows that the proposed definition is a restriction of BLP. Indeed, the new definition is equivalent to transferring the first-level constraints involving second-level variables into the second level, resulting in a special case of BLP in which there are no first-level constraint involving second-level variables. Thus, contrary to what is stated by the authors who suggested the new definition, this does not allow to solve a wider class of problems, but rather relaxes the feasible region, allowing for infeasible points to be considered as feasible.