提出一种无需调参的单循环算法，实现双层优化的实时可行性与上层目标下降。
Sequential QCQP for Bilevel Optimization with Line Search
- 每轮求解可闭式解的凸二次约束二次规划，确定搜索方向。
- 采用受控屏障函数启发的回溯线搜索，保证步长安全且恒正。
- 无需调参、可扩展，适用于多种双层优化任务，收敛性有理论保障。
双层优化具有层次结构，上下层间存在复杂依赖关系。本文提出一种单循环、无需调参的算法，能保证任意时刻的下层近似最优性（即时可行性），同时确保上层目标单调下降。每轮迭代中，通过求解一个具有闭式解的凸二次约束二次规划（QCQP）获得搜索方向，并采用受控屏障函数启发的回溯线搜索，确保步长安全且恒为正。该方法具备可扩展性，无需超参数调节，在较弱的局部正则性假设下收敛。我们建立了关于一阶驻点度量的 O(1/k) 均匀收敛速率，并在典型双层优化任务中验证了其有效性。
原文摘要 · Abstract (English)
Bilevel optimization involves a hierarchical structure where one problem is nested within another, leading to complex interdependencies between levels. We propose a single-loop, tuning-free algorithm that guarantees anytime feasibility, i.e., approximate satisfaction of the lower-level optimality condition, while ensuring descent of the upper-level objective. At each iteration, a convex quadratically-constrained quadratic program (QCQP) with a closed-form solution yields the search direction, followed by a backtracking line search inspired by control barrier functions to ensure safe, uniformly positive step sizes. The resulting method is scalable, requires no hyperparameter tuning, and converges under mild local regularity assumptions. We establish an O(1/k) ergodic convergence rate in terms of a first-order stationary metric and demonstrate the algorithm's effectiveness on representative bilevel tasks.
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