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Duality-based Dynamical Optimal Transport of Discrete Time Systems

Dongjun Wu, Anders Rantzer

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.

Duality-based Dynamical Optimal Transport of Discrete Time Systems

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.

Paper Structure

This paper contains 9 sections, 6 theorems, 62 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Theoretical foundations
  3. Linear Systems with Gaussian Marginals
  4. Splitting algorithm
  5. General problem
  6. A special case
  7. Simulation example
  8. Conclusion
  9. Appendix

Key Result

Theorem 1

Let $\mu_{1}$ and $\mu_{T}$ be two probability measures. Then under Assumption H1-H2: Here we have omitted the arguments $(\mu_{1},\mu_{T})$ for ease of notation. 1) The inequalities hold If $\mu_{1}$ is absolutely continuous, then all the inequalities become equalities. 2) Optimizers exist for Problems pb:K, pb:KR-dual and pb:dual. In particular, if $\{v_{k}\}_{k=1}^{T}$ is a maximizer of Proble

Figures (3)

  • Figure 1: The optimal cost.
  • Figure 2: The evolution of densities.
  • Figure 3: The optimal control sequence $u_{1},\cdots u_{4}$.

Theorems & Definitions (12)

  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1
  • Remark 4
  • Remark 5
  • Proposition 1
  • Theorem 2
  • Corollary 1
  • Remark 6
  • ...and 2 more