On the Complexity of Bilevel Linear and Quadratic Programs in Fixed Dimensions

Abstract

It is well-known that general bilevel linear programs (BLPs) are strongly $NP$-hard, even when the leader's and the follower's objective functions are exact opposites. However, the complexity classification of BLPs remains incomplete when one of the decision-makers has a fixed number of variables or constraints. In this paper, we close the remaining gap in this complexity landscape. Thus, while optimistic BLPs are known to be polynomially solvable when the number of follower variables is fixed, we prove that the corresponding pessimistic problem is strongly $NP$-hard. To the best of our knowledge, this is the first result demonstrating that, under comparable assumptions, the pessimistic formulation can be computationally harder than its optimistic counterpart. In addition, we prove that BLPs remain polynomially solvable in both the optimistic and the pessimistic settings when the number of follower constraints is fixed. We further investigate whether these polynomial-time solvability results persist for bilevel convex quadratic programs. While the optimistic formulation remains polynomially solvable when the number of follower variables is fixed, we prove that the pessimistic formulation with a fixed number of follower constraints becomes $NP$-hard. In other words, unless $P = NP$, there is a strict complexity gap between bilevel programs with linear and convex quadratic objective functions. Finally, we show that replacing a convex quadratic follower objective with a nonconvex quadratic one renders the optimistic problem $NP$-hard, even when both follower dimensions are fixed.