Linear bilevel programming with upper level constraints depending on the lower level solution

Abstract

Focus in the paper is on the definition of linear bilevel programming problems, the existence of optimal solutions and necessary as well as sufficient optimality conditions. In the papers [C. Shi, G. Zhang, J. Lu, An extended Kuhn–Tucker approach for linear bilevel programming, Appl. Math. Comput. 162 (2005) 51–63] and [C. Shi, G. Zhang, J. Lu, On the definition of linear bilevel programming solution, Appl. Math. Comput. 160 (2005) 169–176], the authors claim to suggest a refined definition of linear bilevel programming problems and related optimality conditions. Mainly their attempt reduces to shifting upper level constraints involving both the upper and the lower level variables into the lower level. We investigate such a shift in more details and show that it is not allowed in general. We show that an optimal solution of the bilevel program exists under the conditions in [C. Shi, G. Zhang, J. Lu, On the definition of linear bilevel programming solution, Appl. Math. Comput. 160 (2005) 169–176] if we add the assumption that the inducible region is not empty. The necessary optimality condition reduces to check optimality in one linear programming problem. Optimality of one feasible point for a certain number of linear programs implies optimality for the bilevel problem.