Safe Gradient Flow for Bilevel Optimization

• URL: http://arxiv.org/abs/2501.16520v2
• Date: Fri, 21 Mar 2025 19:49:45 GMT
• Title: Safe Gradient Flow for Bilevel Optimization
• Authors: Sina Sharifi, Nazanin Abolfazli, Erfan Yazdandoost Hamedani, Mahyar Fazlyab,
• Abstract summary: Bilevel optimization is a key framework in hierarchical decision-making.<n>We propose a control-theoretic approach to solving bilevel optimization problems.
• Score: 6.3963012138521975
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Bilevel optimization is a key framework in hierarchical decision-making, where one problem is embedded within the constraints of another. In this work, we propose a control-theoretic approach to solving bilevel optimization problems. Our method consists of two components: a gradient flow mechanism to minimize the upper-level objective and a safety filter to enforce the constraints imposed by the lower-level problem. Together, these components form a safe gradient flow that solves the bilevel problem in a single loop. To improve scalability with respect to the lower-level problem's dimensions, we introduce a relaxed formulation and design a compact variant of the safe gradient flow. This variant minimizes the upper-level objective while ensuring the lower-level decision variable remains within a user-defined suboptimality. Using Lyapunov analysis, we establish convergence guarantees for the dynamics, proving that they converge to a neighborhood of the optimal solution. Numerical experiments further validate the effectiveness of the proposed approaches. Our contributions provide both theoretical insights and practical tools for efficiently solving bilevel optimization problems.

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