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Decentralized and Equitable Optimal Transport

Ivan Lau, Shiqian Ma, César A. Uribe

TL;DR

The D-OT problem is reformulate as a constraint-coupled optimization problem and a single-loop decentralized algorithm with an iteration complexity of $O(1/\epsilon)$ that matches existing centralized first-order approaches is proposed.

Abstract

This paper considers the decentralized (discrete) optimal transport (D-OT) problem. In this setting, a network of agents seeks to design a transportation plan jointly, where the cost function is the sum of privately held costs for each agent. We reformulate the D-OT problem as a constraint-coupled optimization problem and propose a single-loop decentralized algorithm with an iteration complexity of O(1/ε) that matches existing centralized first-order approaches. Moreover, we propose the decentralized equitable optimal transport (DE-OT) problem. In DE-OT, in addition to cooperatively designing a transportation plan that minimizes transportation costs, agents seek to ensure equity in their individual costs. The iteration complexity of the proposed method to solve DE-OT is also O(1/ε). This rate improves existing centralized algorithms, where the best iteration complexity obtained is O(1/ε^2).

Decentralized and Equitable Optimal Transport

TL;DR

The D-OT problem is reformulate as a constraint-coupled optimization problem and a single-loop decentralized algorithm with an iteration complexity of that matches existing centralized first-order approaches is proposed.

Abstract

This paper considers the decentralized (discrete) optimal transport (D-OT) problem. In this setting, a network of agents seeks to design a transportation plan jointly, where the cost function is the sum of privately held costs for each agent. We reformulate the D-OT problem as a constraint-coupled optimization problem and propose a single-loop decentralized algorithm with an iteration complexity of O(1/ε) that matches existing centralized first-order approaches. Moreover, we propose the decentralized equitable optimal transport (DE-OT) problem. In DE-OT, in addition to cooperatively designing a transportation plan that minimizes transportation costs, agents seek to ensure equity in their individual costs. The iteration complexity of the proposed method to solve DE-OT is also O(1/ε). This rate improves existing centralized algorithms, where the best iteration complexity obtained is O(1/ε^2).
Paper Structure (10 sections, 6 theorems, 24 equations, 1 figure, 1 table, 1 algorithm)

This paper contains 10 sections, 6 theorems, 24 equations, 1 figure, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Existing approaches for decentralized and equitable OT problems
  3. Reformulation to DCCO
  4. Reformulating D-OT as a DCCO problem
  5. Reformulating DE-OT as a DCCO problem
  6. DCCO algorithms to solve reformulated OT and EOT
  7. Euclidean regularized D-OT and DE-OT
  8. PDC-ADMM
  9. Numerical simulations
  10. Conclusion

Key Result

Lemma III.1

The OT problem eq: OT is equivalent to where the indicator function $\iota_{\ge \bm{0}}( y)$ is defined by and the matrix $M_i \in \mathbb{R}^{2n \times n}$ is the $i^{\text{th}}$ column-block of

Figures (1)

  • Figure 1: Optimality gap $f(\mathbf{x}^k) - f^*$ and feasibility violation $\|A \mathbf{x}^k - b\|$ for D-OT and DE-OT under Tracking-ADMM and DC-ADMM.

Theorems & Definitions (15)

  • Remark II.1
  • Lemma III.1
  • proof
  • Remark III.2
  • Proposition III.3
  • Proposition III.4
  • proof
  • Proposition III.6
  • Remark III.7
  • Remark IV.1
  • ...and 5 more