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A relaxation viewpoint to Unbalanced Optimal Transport: duality, optimality and Monge formulation

Giuseppe Savaré, Giacomo Enrico Sodini

Abstract

We present a general convex relaxation approach to study a wide class of Unbalanced Optimal Transport problems for finite non-negative measures with possibly different masses. These are obtained as the lower semicontinuous and convex envelope of a cost for non-negative Dirac masses. New general primal-dual formulations, optimality conditions, and metric-topological properties are carefully studied and discussed.

A relaxation viewpoint to Unbalanced Optimal Transport: duality, optimality and Monge formulation

Abstract

We present a general convex relaxation approach to study a wide class of Unbalanced Optimal Transport problems for finite non-negative measures with possibly different masses. These are obtained as the lower semicontinuous and convex envelope of a cost for non-negative Dirac masses. New general primal-dual formulations, optimality conditions, and metric-topological properties are carefully studied and discussed.
Paper Structure (20 sections, 39 theorems, 379 equations)

This paper contains 20 sections, 39 theorems, 379 equations.

Table of Contents

  1. Introduction
  2. Plan of the paper.
  3. Acknowledgments.
  4. Preliminaries, measures and functions on the cone
  5. The cone construction
  6. Functions on the product cone
  7. Examples of cost functions
  8. Mass-space product costs
  9. Homogeneous marginal perspective functional
  10. Convexification and duality
  11. Unbalanced Optimal Transport as convex relaxation of singular cost on Dirac masses
  12. Disintegration, barycentric projection and decomposition of homogeneous couplings
  13. The dual problem
  14. The Monge formulation
  15. Existence of a maximazing pair for the dual problem
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $\mathsf X_1, \mathsf X_2$ be completely regular spaces and let $\mathsf H:\mathfrak{C}[\mathsf X_1] \times \mathfrak{C}[\mathsf X_2] \to [0, + \infty]$ be a proper, radially $1$-homogeneous and lower semicontinuous function.

Theorems & Definitions (98)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Remark 2.5
  • ...and 88 more