The goal is to optimize the blend of formulations

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Unformatted text preview: moved directly from node j to node i.) Let Sij ≥ 0 denote the amount of earth moved from location j to location i. The total cost is n =1 Sij Cij = tr C T S . The shipment matrix S must satisfy the balance equations, i,j n n Sij = yi , Sij = xj , i = 1, . . . , n, j = 1, . . . , n, i=1 j =1 which we can write compactly as S 1 = y , S T 1 = x. (The first equation states that the total amount shipped into location i equals yi ; the second equation states that the total shipped out from location j is xj .) The earth mover’s distance between x and y , denoted d(x, y ), is given by the minimal cost of earth moving required to transform x to y , i.e., the optimal value of the problem minimize tr C T S subject to Sij ≥ 0, S 1 = y, i, j = 1, . . . , n S T 1 = x, with variables S ∈ Rn×n . We can also give a probability interpretation of d(x, y ). Consider a random variable Z on {1, . . . , n}2 with values Cij . We seek the joint distribution S that minimizes the expected value of the random variable Z , with given marginals x and y . The earth move...
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This note was uploaded on 09/10/2013 for the course C 231 taught by Professor F.borrelli during the Fall '13 term at Berkeley.

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