Overlapping Schwarz Scheme for Linear-Quadratic Programs in Continuous Time

2025-10-06

AI Summary

This paper addresses the efficient continuous-time solution of time-inhomogeneous linear-quadratic optimal control problems (OCPs). To solve the associated Hamiltonian two-point boundary-value problem, we propose a “optimize-then-discretize” overlapping Schwarz parallel algorithm: the time domain is partitioned with overlap; boundary condition updates are derived from Pontryagin’s minimum principle; and, for the first time, the exponential sensitivity decay property—previously established only in discrete time—is extended to the continuous-time setting to ensure global convergence. The method accommodates diverse numerical integrators, including adaptive-step ODE solvers, without requiring prior discretization of the underlying ordinary differential equation model. Numerical experiments demonstrate high parallel efficiency and robust convergence across multiple scientific computing benchmarks, significantly enhancing the scalability and practicality of solving large-scale continuous-time OCPs.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Machine Learning: Optimization

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We present an optimize-then-discretize framework for solving linear-quadratic optimal control problems (OCP) governed by time-inhomogeneous ordinary differential equations (ODEs). Our method employs a modified overlapping Schwarz decomposition based on the Pontryagin Minimum Principle, partitioning the temporal domain into overlapping intervals and independently solving Hamiltonian systems in continuous time. We demonstrate that the convergence is ensured by appropriately updating the boundary conditions of the individual Hamiltonian dynamics. The cornerstone of our analysis is to prove that the exponential decay of sensitivity (EDS) exhibited in discrete-time OCPs carries over to the continuous-time setting. Unlike the discretize-then-optimize approach, our method can flexibly incorporate different numerical integration methods for solving the resulting Hamiltonian two-point boundary-value subproblems, including adaptive-time integrators. A numerical experiment on a linear-quadratic OCP illustrates the practicality of our approach in broad scientific applications.

Problem

• Solving linear-quadratic optimal control problems with time-inhomogeneous ODEs
• Developing overlapping Schwarz decomposition using Pontryagin Minimum Principle
• Ensuring convergence through boundary condition updates in Hamiltonian systems

Innovation

• Optimize-then-discretize framework for linear-quadratic control problems
• Modified overlapping Schwarz decomposition with Pontryagin Principle
• Flexible numerical integration methods for Hamiltonian subproblems