A Primal-dual Hybrid Gradient Method for Optimal Control Problems and Hamilton-Jacobi PDEs

Core Concepts

The authors introduce an innovative optimization-based approach using the preconditioned primal-dual hybrid gradient method to solve optimal control problems and Hamilton-Jacobi PDEs efficiently.

Abstract

The content discusses a novel optimization-based approach utilizing the preconditioned primal-dual hybrid gradient method to address optimal control problems and Hamilton-Jacobi partial differential equations. The method reformulates these problems into a saddle point problem, enabling efficient solutions with first-order accuracy and numerical stability. The framework extends to viscous HJ PDEs and stochastic optimal control problems, showcasing versatility in managing diverse Hamiltonians. Through numerical examples, the effectiveness of the method is demonstrated in handling various scenarios, highlighting its potential for broad applications.

Stats

Tingwei Meng et al. are partially funded by Air Force Office of Scientific Research (AFOSR) MURI FA9550-18-502 and Office of Naval Research (ONR) N00014-20-1-2787.
Wuchen Li is supported by multiple grants from different organizations including Air Force Office of Scientific Research (AFOSR), National Science Foundation (NSF), among others.

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A Primal-dual hybrid gradient method for solving optimal control problems and the corresponding Hamilton-Jacobi PDEs

by Tingwei Meng... at arxiv.org 03-06-2024

https://arxiv.org/pdf/2403.02468.pdf

Deeper Inquiries

How does the proposed optimization-based approach compare to traditional methods in solving optimal control problems

The proposed optimization-based approach offers several advantages over traditional methods in solving optimal control problems. Firstly, it reformulates the optimal control problem and Hamilton-Jacobi PDEs into a saddle point problem using a Lagrange multiplier, which allows for more efficient and effective optimization. This approach enables the use of the preconditioned primal-dual hybrid gradient (PDHG) method, which provides solutions with first-order accuracy and numerical unconditional stability. Compared to traditional grid-based methods or explicit time discretization schemes, this methodology allows for larger time steps and avoids limitations such as the Courant–Friedrichs–Lewy (CFL) condition.

What are the implications of handling non-smooth Hamiltonians in the context of optimal control using this methodology

Handling non-smooth Hamiltonians in the context of optimal control using this methodology has significant implications. The proposed approach can effectively handle a wide variety of Hamiltonian functions, including those that exhibit non-smooth behavior and dependencies on both time and space variables. By formulating these challenges into a saddle point problem with appropriate updates for ρ, α1, α2 (or α11, α12, α21, α22 in two-dimensional cases), the algorithm can efficiently compute solutions even when dealing with complex or irregular Hamiltonians. This capability is crucial for addressing real-world problems where smoothness assumptions may not hold.

How can the findings from this study be applied to real-world scenarios beyond mathematics

The findings from this study have broad applications beyond mathematics in various real-world scenarios. For instance:

In robotics: The methodology can be applied to trajectory planning for robots or manipulators by optimizing their paths while considering constraints.
In autonomous vehicles: It can help improve decision-making processes by finding optimal controls based on dynamic environments.
In finance: The framework could be utilized to optimize investment strategies under uncertain market conditions.
In healthcare: It might assist in designing personalized treatment plans by optimizing drug dosages or therapy schedules based on individual patient data.
Overall, the versatility and efficiency of this optimization-based approach make it applicable across diverse fields where optimal control plays a critical role in decision-making processes.