A Dantzig-Wolfe Single-Level Reformulation for Mixed-Integer Linear Bilevel Optimization: Exact and Heuristic Approaches

Bilevel optimization problems arise in numerous real-world applications. While single-level reformulations are a common strategy for solving convex bilevel problems, such approaches usually fail when the follower’s problem includes integer variables. In this paper, we present the first single-level reformulation for mixed-integer linear bilevel optimization, which does not rely on the follower’s value function. Our approach is based on convexifying the follower’s problem via a Dantzig-Wolfe reformulation and exploits strong duality of the reformulated problem. By doing so, we derive a nonlinear single-level problem, which is equivalent to the original bilevel model. Moreover, we show that this problem can be transformed into a mixed-integer linear problem using standard linearization techniques and bounds on the dual variables of the convexified follower’s problem. Notably, we show that these bounds can be computed in practice via a polynomial-time-solvable problem, which is purely based on the primal problem’s data. This results in a new branch-and-cut approach for mixed-integer linear bilevel optimization. In addition to this exact solution approach, we also present a penalty alternating direction method, which computes high-quality feasible points. Numerical experiments on instances from the BOBILib show that we are able find feasible points for 66 instances that were still open and to improve the quality of the best known solution for 12 instances.

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