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arXiv · 2409.06530

Functionally Constrained Algorithm Solves Convex Simple Bilevel Problems

Huaqing Zhang·Lesi Chen·Jing Xu·Jingzhao Zhang
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Abstract

This paper studies simple bilevel problems, where a convex upper-level function is minimized over the optimal solutions of a convex lower-level problem. We first show the fundamental difficulty of simple bilevel problems, that the approximate optimal value of such problems is not obtainable by first-order zero-respecting algorithms. Then we follow recent works to pursue the weak approximate solutions. For this goal, we propose a novel method by reformulating them into functionally constrained problems. Our method achieves near-optimal rates for both smooth and nonsmooth problems. To the best of our knowledge, this is the first near-optimal algorithm that works under standard assumptions of smoothness or Lipschitz continuity for the objective functions.

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Huaqing Zhang, Lesi Chen, Jing Xu, Jingzhao Zhang. 2024-09-10. Functionally Constrained Algorithm Solves Convex Simple Bilevel Problems. https://arxiv.org/abs/2409.06530

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