A true single-level reformulation for pessimistic bilevel optimization

Abstract: We propose a single-level reformulation (SLR) for the pessimistic bilevel optimization problem that does not rely on complementarity conditions or optimal value functions. For this reason, we refer to it as a true single-level reformulation (tSLR). A remarkable consequence is that this reformulation can satisfy the classical linear independence constraint qualification, despite the fact that even the weaker Mangasarian-Fromovitz constraint qualification is known to systematically fail for standard single-level reformulations of both optimistic and pessimistic bilevel programs. We leverage on these constraint qualifications to construct new necessary optimality conditions for pessimistic bilevel optimization. The reformulation also has a striking limitation: under the assumptions of our analysis, the classical second-order sufficient condition fails at every Karush-Kuhn-Tucker point of the problem. Nevertheless, preliminary numerical experiments demonstrate that algorithms based on the proposed tSLR can outperform existing approaches for pessimistic bilevel optimization. Overall, the proposed framework suggests that pessimistic bilevel programs may be considerably more tractable than previously believed and need not be inherently more difficult to solve than their optimistic counterparts.