A Single-Loop First-Order Algorithm for Linearly Constrained Bilevel Optimization

We study bilevel optimization problems where the lower-level problems are strongly convex and have coupled linear constraints. To overcome the potential non-smoothness of the hyper-objective and the computational challenges associated with the Hessian matrix, we utilize penalty and augmented Lagrangian methods to reformulate the original problem as a single-level one. Especially, we establish a strong theoretical connection between the reformulated function and the original hyper-objective by characterizing the closeness of their values and derivatives. Based on this reformulation, we propose a single-loop, first-order algorithm for linearly constrained bilevel optimization (SFLCB). We provide rigorous analyses of its non-asymptotic convergence rates, showing an improvement over prior double-loop algorithms – form $O(ε^{-3}\log(ε^{-1}))$ to $O(ε^{-3})$. The experiments corroborate our theoretical findings and demonstrate the practical efficiency of the proposed SFLCB algorithm. Simulation code is provided at https://github.com/ShenGroup/SFLCB.

💡 Research Summary

The paper addresses bilevel optimization (BLO) problems in which the lower‑level (LL) subproblem is strongly convex in the variable y and is subject to coupled linear constraints of the form h(x, y)=Bx+Ay−b≤0. The upper‑level (UL) objective f(x, y) and the LL objective g(x, y) are assumed to be continuously differentiable. Traditional approaches for constrained BLO either rely on implicit gradient methods that require Hessian‑vector products of the LL problem or employ double‑ or triple‑loop schemes that lead to sub‑optimal convergence rates and cumbersome implementations.

The authors first reformulate the original bilevel problem into a single‑level penalized problem using a combination of penalty and augmented Lagrangian techniques. Specifically, they define a penalized function

 Φ₍δ₎(x)=min_{y∈Y(x)} max_{z∈Y(x)}