A Single-Loop Bilevel Deep Learning Method for Optimal Control of Obstacle Problems

2026-01-07

AI Summary

This work addresses the computational challenges posed by nonsmoothness, nonlinearity, and bilevel structure in optimal control problems governed by obstacle equations. The authors propose a single-loop bilevel deep learning framework that employs mesh-free neural networks to simultaneously approximate both state and control variables, preserving the original bilevel formulation while circumventing repeated solves of discretized subproblems. Central to this approach is the novel single-loop stochastic first-order bilevel algorithm, S2-FOBA, which eliminates the need for nested optimization and does not rely on the uniqueness of lower-level solutions, thereby enabling applicability to high-dimensional problems over complex domains. Numerical experiments on benchmark obstacle control problems demonstrate that the method achieves accuracy comparable to or better than conventional numerical schemes at substantially reduced computational cost.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Intelligent Robots: Learning & Optimization for ROB
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Optimal control of obstacle problems arises in a wide range of applications and is computationally challenging due to its nonsmoothness, nonlinearity, and bilevel structure. Classical numerical approaches rely on mesh-based discretization and typically require solving a sequence of costly subproblems. In this work, we propose a single-loop bilevel deep learning method, which is mesh-free, scalable to high-dimensional and complex domains, and avoids repeated solution of discretized subproblems. The method employs constraint-embedding neural networks to approximate the state and control and preserves the bilevel structure. To train the neural networks efficiently, we propose a Single-Loop Stochastic First-Order Bilevel Algorithm (S2-FOBA), which eliminates nested optimization and does not rely on restrictive lower-level uniqueness assumptions. We analyze the convergence behavior of S2-FOBA under mild assumptions. Numerical experiments on benchmark examples, including distributed and obstacle control problems with regular and irregular obstacles on complex domains, demonstrate that the proposed method achieves satisfactory accuracy while reducing computational cost compared to classical numerical methods.

Problem

• optimal control
• obstacle problems
• bilevel optimization
• nonsmoothness
• nonlinearity

Innovation

• bilevel optimization
• mesh-free deep learning
• obstacle problems
• single-loop algorithm
• constraint-embedding neural networks