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An optimal matching problem. (English) Zbl 1106.49054

The problem considered in the paper deals with optimal mass transportation theory and has an application in some economic theories of pricing. The mathematical formulation of the problem is as follows: two sets \(X\) and \(Y\) are given, with positive measures \(\mu\) and \(\nu\) such that \(\mu(X)=\nu(Y)\); given a third set \(Z\) we want to find transport maps \(s:X\to Z\) and \(t:Y\to Z\) that satisfy the “matching condition” \(s^\#(\mu)=t^\#(\nu)\), where \(\#\) denotes the push-forward operation, and that minimize the total transportation cost \[ \int_Xw\bigl(x,s(x)\bigr)\,d\mu+\int_Yv\bigl(y,t(y)\bigr) \,d\nu, \] where \(w\) and \(v\) are two given cost functions. The dual approach of Kantorovich is followed, by introducing a new problem whose solution provides the solution of the original one. The last section of the paper contains some explicit examples.

MSC:

49Q20 Variational problems in a geometric measure-theoretic setting
05A15 Exact enumeration problems, generating functions
05C38 Paths and cycles
91B28 Finance etc. (MSC2000)
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References:

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