An Exact Algorithm for Mixed-Integer Bilevel Stochastic Problem

Tomás Lagos; Dmytro Matsypura

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arXiv · 2606.29344

An Exact Algorithm for Mixed-Integer Bilevel Stochastic Problem

Tomás Lagos·Dmytro Matsypura

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Abstract

We study a class of mixed-integer bilevel stochastic programs in which the leader commits to a first-stage decision before uncertainty is realized, and the follower solves a mixed-integer optimization problem for each revealed scenario. Due to the hierarchical structure and discrete variables at both levels, these problems are inherently $\Sigma_2^p$-hard, rendering standard single-level reformulations computationally intractable. To address this challenge, we develop an exact algorithm that combines deterministic value-function reformulations with 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 and updates the Lagrange multipliers. We prove that, under boundedness assumptions, our algorithm converges in finite time to a global optimum and provides valid upper and lower bounds throughout its execution.

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Tomás Lagos, Dmytro Matsypura. 2026-06-28. An Exact Algorithm for Mixed-Integer Bilevel Stochastic Problem. https://arxiv.org/abs/2606.29344

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