Benders Subproblem Decomposition for Bilevel Problems with Convex Follower

Abstract

Bilevel optimization formulates hierarchical decision-making processes that arise in many real-world applications such as in pricing, network design, and infrastructure defense planning. In this paper, we consider a class of bilevel optimization problems where the upper-level problem features some integer variables while the lower-level problem enjoys strong duality. We propose a dedicated Benders decomposition method for solving this class of bilevel problems, which decomposes the Benders subproblem into two more tractable, sequentially solvable problems that can be interpreted as the upper and the lower-level problems. We show that the Benders subproblem decomposition carries over to an interesting extension of bilevel problems, which connects the upper-level solution with the lower level dual solution, and discuss some special cases of bilevel problems that allow sequence-independent subproblem decomposition. Several novel schemes for generating numerically stable cuts, finding a good incumbent solution, and accelerating the search tree are discussed. A computational study demonstrates the computational benefits of the proposed method over a state-of-the-art bilevel-tailored branch-and-cut method, a commercial solver, and the standard Benders method on standard test cases and the motivating applications in sequential energy markets.