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Duality-based Dynamical Optimal Transport of Discrete Time Systems

Wu, Dongjun LU and Rantzer, Anders LU orcid (2024)
Abstract
We study dynamical optimal transport of discrete time systems (dDOT) with Lagrangian cost. The problem is approached by combining optimal control and Kantorovich duality theory. Based on the derived solution, a first order splitting algorithm is proposed for numerical implementation. While solving partial differential equations is often required in the continuous time case, a salient feature of our algorithm is that it avoids equation solving entirely. Furthermore, it is typical to solve a convex optimization problem at each grid point in continuous time settings, the discrete case reduces this to a straightforward maximization. Additionally, the proposed algorithm is highly amenable to parallelization. For linear systems with Gaussian... (More)
We study dynamical optimal transport of discrete time systems (dDOT) with Lagrangian cost. The problem is approached by combining optimal control and Kantorovich duality theory. Based on the derived solution, a first order splitting algorithm is proposed for numerical implementation. While solving partial differential equations is often required in the continuous time case, a salient feature of our algorithm is that it avoids equation solving entirely. Furthermore, it is typical to solve a convex optimization problem at each grid point in continuous time settings, the discrete case reduces this to a straightforward maximization. Additionally, the proposed algorithm is highly amenable to parallelization. For linear systems with Gaussian marginals, we provide a semi-definite programming formulation based on our theory. Finally, we validate the approach with a simulation example. (Less)
Please use this url to cite or link to this publication:
author
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organization
publishing date
type
Working paper/Preprint
publication status
submitted
subject
project
Scalable Control of Interconnected Systems
language
English
LU publication?
yes
id
9a69150d-8945-4677-b243-a4202004e21c
alternative location
https://arxiv.org/abs/2410.09801
date added to LUP
2025-01-08 16:49:51
date last changed
2025-04-04 14:30:39
@misc{9a69150d-8945-4677-b243-a4202004e21c,
  abstract     = {{We study dynamical optimal transport of discrete time systems (dDOT) with Lagrangian cost. The problem is approached by combining optimal control and Kantorovich duality theory. Based on the derived solution, a first order splitting algorithm is proposed for numerical implementation. While solving partial differential equations is often required in the continuous time case, a salient feature of our algorithm is that it avoids equation solving entirely. Furthermore, it is typical to solve a convex optimization problem at each grid point in continuous time settings, the discrete case reduces this to a straightforward maximization. Additionally, the proposed algorithm is highly amenable to parallelization. For linear systems with Gaussian marginals, we provide a semi-definite programming formulation based on our theory. Finally, we validate the approach with a simulation example.}},
  author       = {{Wu, Dongjun and Rantzer, Anders}},
  language     = {{eng}},
  note         = {{Preprint}},
  title        = {{Duality-based Dynamical Optimal Transport of Discrete Time Systems}},
  url          = {{https://arxiv.org/abs/2410.09801}},
  year         = {{2024}},
}