OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...

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Example TakeN∑N∑f + = δ pi f − = δ ni .i=1i=1Then the optimal Monge cost is given by theminimal connection of the p i with the n i .Relaxed formulation (due to Kantorovich):consider measures γ on X × X• γ is an admissible transport plan ifπ # 1 γ = f + and π # 2 γ = f − .• Transport plans γ that are concentratedon γ-measurable graphs are actually transportmaps T : X → X, via the equalityγ = (Id × T ) # f + .5
Monge-Kantorovich problem:{ ∫ }min d(x, y) dγ(x, y) : γ admiss. .X×XTheorem There exists an optimal transportplan γ opt ; in the Euclidean case γ optis actually a transport map T opt wheneverf + and f − are in L 1 .The existence part is easy; indeed in thecase X compact it follows from the weak*compactness of probabilities and the weak*continuity of the Monge-Kantorovich cost.In the general case the same holds by usingthe boundedness of the first moments.Note that, while in the Monge problem thecost is highly nonlinear with respect to theunknown T , in the Kantorovich formulationthe cost is linear with respect to the unknownγ.6
Page 1 and 2: OPTIMIZATION PROBLEMS INMASS TRANSP
Page 3 and 4: The Monge problem is thenmin{ ∫ d
Page 5: Then the class of admissible transp
Page 9 and 10: Shape optimization problemsThere is
Page 12 and 13: The shape optimization problem abov
Page 14 and 15: To overcome some of these difficult
Page 16 and 17: In the mass optimization problem th
Page 18 and 19: More precisely, we consider the geo
Page 20 and 21: produces an optimal µ which has a
Page 22 and 23: Variational integrals w.r.t. a meas
Page 24 and 25: Here are some cases where the optim
Page 26 and 27: -2 -1.5 -1 -0.5 0 0.5 1 1.5 21.510.
Page 28 and 29: Problem 1 - Optimal NetworksWe cons
Page 30 and 31: •Large town policy: several ticke
Page 32 and 33: It is interesting to study the opti
Page 34 and 35: The function B can be seen as a con
Page 36 and 37: Here is another case, with a single
Page 38 and 39: • there is a transportation cost
Page 40 and 41: We have then the optimization probl
Page 42 and 43: and so the value of the cost F (a).
Page 44 and 45: is a delicate matter. Indeed, maxim
Page 46 and 47: Some Numerical ComputationsHere are
Page 48 and 49: Optimal sets of length 0.5 and 1 in
Page 50 and 51: Now we give a model that takes into
Page 52 and 53: We also give a model where the oppo
Page 54 and 55: Theorem Assume:• J is lower semic
Page 56 and 57:
∫∫ ∫J(µ) = f(µ a )dm+ f ∞
Page 58 and 59:
Concentration case. The following f
Page 60 and 61:
Diffusion case. The following facts
Page 62 and 63:
Theorem Let (X, d, d ′ ) be a met
Page 64 and 65:
• if q < 1 + 1/N for every µ 0 a
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OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...

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