An Exact Algorithm for Mixed-Integer Bilevel Stochastic Problem

Abstract

We study a class of mixed-integer bilevel stochastic programs where the leader commits to a first-stage decision before uncertainty is realized, and the follower solves a subsequent mixed-integer optimization problem for each revealed scenario. Due to the hierarchical structure and the presence of discrete variables at both levels, these problems are inherently $Σ_2^p$-hard, making standard single-level reformulations computationally intractable. To address this significant computational challenge, we develop an exact algorithm that combines deterministic value-function reformulations with stochastic scenario-wise decomposition. Specifically, we propose an extended single-level reformulation and a corresponding relaxation that enable scenario decomposition. We then introduce a stochastic subgradient cutting-plane scheme that dynamically generates follower optimality cuts while updating the Lagrange multipliers. We prove that, under boundedness assumptions, our algorithm converges in finite time to a true global optimum while providing valid upper and lower bounds throughout its execution.