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- W2186118494 abstract "This Master Project presents some basic aspects of the Monge–Kantorovich problem, also known in the literature as the “mass transportation problem”. There are two goals for this project. The first is to survey theoretical ideas, such as existence and uniqueness proofs and characterization of the solutions. The second is to deal with the practical question of explicitly finding the solutions for two families of problems: the discrete ones and the absolutely continuous ones. In the introduction, the problem is formulated in measure theoretical terms as an optimization problem over measures, and examples are given. In Section 2 a dual problem for the Monge–Kantorovich problem, involving optimization over functions, is surveyed and a duality theorem is given. In the particular case where a finite number of points is to be transported to equally many locations, the problem can be recast as a linear program; the foundations of the latter is the topic of Section 3. In Section 4 an efficient algorithm for this linear program, the Hungarian method, is presented. Section 5 deals with absolutely continuous measures in Euclidean spaces. The particular cost function, squared Euclidean distance, gives rise to an algorithm for finding the optimal transportation; this is sketched in Section 6. In the last section, we return to the case of arbitrary measures, and optimality is characterized by a property called c-monotonicity. An elegant stability result of the Monge–Kantorovich problem is established as a corollary. This is seen to provide, also, a heuristic for interpreting transportation of absolutely continuous measures via the Hungarian method. The image to the left in the title page is from http://de.wikipedia.org/w/index. php?title=Datei:SBB_482_Kesselwagen.jpg&filetimestamp=20090717124221 and is attributed to the Wikipedia user Niesen. The second image is from http:// upload.wikimedia.org/wikipedia/commons/e/e6/Lego-Chicago_Rush_Hour.jpg, and is attributed to the user Joe. Both were retrieved June 22, 2012." @default.
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- W2186118494 date "2012-01-01" @default.
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- W2186118494 title "Optimal Transportation: Continuous and Discrete" @default.
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