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

2023/12/31 by Giuseppe Savaré, Savaré, Giuseppe, Giacomo Enrico Sodini +1 · 1 citation
Mathematics · #28A33 #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations #Optimization and Control (math.OC) #Primary: 49Q22 #Secondary: 49K27

paper · pdf · doi:10.48550/arxiv.2401.00542

openalex publication_date 2023/12/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Cited by

• Benamou-Brenier and Kantorovich on sub-Riemannian manifolds with no abnormal geodesics

Related

• Introduction to Optimal Transport Theory
• Convex duality in nonlinear optimal transport
• Weak Optimal Transport: When is the Dual Potential Convex?
• Regularity theory and geometry of unbalanced optimal transport
• Nonlinear Dynamical Unbalanced Optimal Transport: Relaxation and Duality