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The Optimal Transport Problem

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

• Kantorovich Problem; Villani (1-3, 6-9), Santambrogio (xv-xvii,1-9)
• 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 and the Monge Ampère equation; Santambrogio (xvi, 54-57)
• Optimal Transport in One Dimension; Villani (73-78); Santambrogio (59-67)

Numerical Methods for Optimal Transport

• Discrete Optimal Transport; Villani (5), Santambrogio (235-237), Peyré Cuturi (7-12)
• Entropic Regularization; Santambrogio (240-241); Peyré Cuturi (57-62)
• Sinkhorn's Algorithm; Peyré Cuturi (62-73)
• 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: Functional Analysis

• The dual of $C_{o}(X)$ vs. $C_{b}(X)$; Villani (39-43); Santambrogio (4); Rudin Real and Complex Analysis (127-132)
• Convergence of Measures and Metrizability (click on link for references and explanation)

Mathematical Foundations: Optimization

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