The classical Optimal Transport (OT) problem studies how to transport one distribution to another in the most efficient way. In the past few decades it has emerged as a very powerful tool in various fields, such as optimization theory, probability theory, partial differential equations, machine learning and data analysis. In this thesis, we will discuss some existing variants of the classical optimal transport problem, such as the capacity constrained OT problem, multi-marginal OT problem, entropy-regularized OT problem and barycenters, and we will introduce a couple of new variants by combining the existing versions. We will also discuss their duality results and some characterizations.
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