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- W2138305655 abstract "Optimal transportation with capacity constraints, a variant of the well-known optimal transportation problem, is concerned with transporting one probability density $$f in L^1(mathbf {R}^m)$$ onto another one $$g in L^1(mathbf {R}^n)$$ so as to optimize a cost function $$c in L^1(mathbf {R}^{m+n})$$ while respecting the capacity constraints $$0le h le bar{h}in L^infty (mathbf {R}^{m+n})$$ . A linear programming duality for this problem was first proposed by Levin. In this note, we prove under mild assumptions on the given data, the existence of a pair of $$L^1$$ -functions optimizing the dual problem. Using these functions, which can be viewed as Lagrange multipliers to the marginal constraints $$f$$ and $$g$$ , we characterize the solution $$h$$ of the primal problem. We expect these potentials to play a key role in any further analysis of $$h$$ . Moreover, starting from Levin’s duality, we derive the classical Kantorovich duality for unconstrained optimal transport. In tandem with results obtained in our companion paper [Korman et al. J Convex Anal arXiv:1309.3022 [8] (in press)], this amounts to a new and elementary proof of Kantorovich’s duality." @default.
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- W2138305655 date "2014-11-09" @default.
- W2138305655 modified "2023-10-14" @default.
- W2138305655 title "Dual potentials for capacity constrained optimal transport" @default.
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- W2138305655 doi "https://doi.org/10.1007/s00526-014-0795-9" @default.
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