A Barrier-Based First-Order Method for Constrained Bilevel Optimization
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ge-gec_mike-sm-aeroastro-2026-thesis.pdf
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Author(s)
Ge, Cheng
Advisor(s)
Jadbabaie, Ali
Date Issued
February 2026
Publisher
Massachusetts Institute of Technology
Abstract
Bilevel optimization (BLO) problems find wide-ranging applications in network optimization, transportation, game theory, and machine learning. In these problems, the upper-level (UL) objective depends on both the UL decision variable and the optimal solution of a lower-level (LL) problem. While recent work has focused on developing scalable first-order methods for BLO, the presence of LL constraints poses significant challenges for first-order gradient-based approaches. In this thesis, we study constrained BLO problems through barrier reformulation, where the LL constraints are incorporated into the LL objective via log-barrier functions. We propose a first-order algorithm for solving barrier-reformulated constrained BLO problems and establish convergence rates of O˜(ϵ −3 t −5 ) when the gradient of the UL objective is subject to stochastic noise, and O˜(ϵ −2 t −5 ) in deterministic settings, where ϵ denotes the level of approximate stationarity and t is the barrier coefficient. These rates match existing lower bounds for nonconvex first-order optimization, up to logarithmic factors in their dependence on ϵ. We also validate the effectiveness of the proposed method in real applications including price setting and hyperparameter optimization. Furthermore, for the special case where the LL objective is quadratic, and the LL constraints are linear, we establish how to choose the barrier coefficient t to ensure that an ϵ-approximate stationary point of the original constrained BLO problem corresponds to an ϵ-approximate stationary point of the barrier-reformulated problem.
MIT Department
Massachusetts Institute of Technology. Department of Aeronautics and Astronautics
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