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We study a parameter (σ)
dependent relaxation of the Travelling Salesman Problem on .
The relaxed problem is reduced to the Travelling Salesman Problem
as 0. For increasing σ it is also an
ordered clustering algorithm for a set of points in .
A dual formulation is introduced, which reduces the problem to a
convex optimization, provided the minimizer is in the domain of
convexity of the relaxed functional. It is shown that this last
condition is generically satisfied, provided σ is large
enough.
...
Let X and Y be two compact spaces endowed with
respective measures μ and ν satisfying the condition µ(X) = v(Y). Let c be a continuous function on the product space X x Y. The mass transfer problem consists in determining a measure ξ on
X x Y whose marginals coincide with μ and ν, and such that
the total cost ∫ ∫ c(x,y)dξ(x,y) be minimized. We first
show that if the cost function c is decomposable, i.e., can be
represented as the sum of two continuous functions defined on X and
Y, respectively,...
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