On Coupling Constraints in Pessimistic Linear Bilevel Optimization
Dorothee Henke, Henri Lefebvre, Martin Schmidt, Johannes Thürauf
TL;DR
The paper tackles pessimistic linear bilevel optimization with coupling constraints and shows that such problems exert no loss in generality compared to their no-coupling counterparts: a pessimistic problem with coupling constraints can be reformulated into a pessimistic problem without coupling constraints that preserves all global optima, and can also be transformed into an optimistic problem without coupling constraints with identical optima. The approach combines a component-wise reformulation of coupling constraints across the follower’s optimal set $S(x)$ (via results from $\text{Zeng}(2020)$) and a penalty-based epigraph technique (via $\text{Henke et al.}(2024)$ and related work), yielding polynomial-sized reformulations. This enables transferring theory and solution techniques between pessimistic and optimistic bilevel models and across coupling/no-coupling settings, challenging the belief that coupling constraints inherently increase difficulty in pessimistic problems. The results broaden modeling flexibility and provide a unified framework for solving linear bilevel problems by moving to equivalent no-CC formulations or to optimistic no-CC formulations, where existing methods may be leveraged.
Abstract
The literature on pessimistic bilevel optimization with coupling constraints is rather scarce and it has been common sense that these problems are harder to tackle than pessimistic bilevel problems without coupling constraints. In this note, we show that this is not the case. To this end, given a pessimistic problem with coupling constraints, we derive a pessimistic problem without coupling constraints that has the same set of globally optimal solutions. Moreover, our results also show that one can equivalently replace a pessimistic problem with such constraints with an optimistic problem without coupling constraints. This paves the way of both transferring theory and solution techniques from any type of these problems to any other one.