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The Optimal Transport Problem

Unless otherwise specified, all topics are for general cost functions c(x,y).

• Kantorovich Dual Problem (for general costs); Villani (17-21), Santambrogio (9-16)
• Kantorovich Dual Problem (for $c(x,y)=d(x,y)$ where $d$ is a metric); Villani (34)
• Kantorovich Dual Problem (for $c(x,y)=d(x,y)^{2}$ where $d$ is a metric); Santambrogio (16-18)
• Optimal Transport in One Dimension; Villani (73-78); Santambrogio (59-67)
• 1-Wasserstein metric, duality, and measures with unequal mass; Piccoli, Rossi, and Tournus

Numerical Methods for Optimal Transport

• Semidiscrete Optimal Transport (for $c(x,y)=|x-y|^{2}$); Santambrogio (242-248); Peyré Cuturi (85-89)

Applications of Optimal Transport

• Machine Learning Kolouri, et al, Optimal Mass Transport: Signal processing and machine-learning applications
• Economic Matching Problems; Santambrogio (44-48) Galichon, A survey of some recent applications of optimal transport methods to econometrics

Mathematical Foundations: Optimization

• Fenchel-Rockafellar and Linear Programming; Brezis (15-17); Rockafellar, Variational Analysis (505-507)