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Optimal partial transport and constrained matching problems

Abstract : The manuscript deals with the mathematical and numerical analysis of the optimal partial transport and optimal constrained matching problems. These two problems bring out new unknown quantities, called active submeasures. For the optimal partial transport with Finsler distance costs, we introduce equivalent formulations characterizing active submeasures, Kantorovich potential and optimal flow. In particular, the PDE of optimality condition allows to show the uniqueness of active submeasures. We then study in detail numerical approximations for which the convergence of discretization and numerical simulations are provided. For Lagrangian costs, we derive and justify rigorously characterizations of solution as well as equivalent formulations. Numerical examples are also given. The rest of the thesis presents the study of the optimal constrained matching with the Euclidean distance cost. This problem has a different behaviour compared to the partial transport. The uniqueness of solution and equivalent formulations are studied under geometric condition. The convergence of discretization and numerical examples are also indicated. The main tools which we use in the thesis are some combinations of PDE techniques, optimal transport theory and Fenchel--Rockafellar dual theory. For numerical computation, we make use of augmented Lagrangian methods.
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Submitted on : Tuesday, November 21, 2017 - 6:22:20 PM
Last modification on : Friday, January 7, 2022 - 3:48:58 AM

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  • HAL Id : tel-01643957, version 1

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van Thanh Nguyen. Optimal partial transport and constrained matching problems. Optimization and Control [math.OC]. Université de Limoges, 2017. English. ⟨NNT : 2017LIMO0052⟩. ⟨tel-01643957⟩

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