An optimal transportation problem with import/export taxes on the boundary

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1 An optimal transportation problem with import/export taxes on the boundary Julián Toledo Workshop International sur les Mathématiques et l Environnement Essaouira, November Joint work with J. M. Mazón and J. D. Rossi

2 Schedule 1. Statement of the problem 2. Mass Transport Theory 3. The mass transport problem with import/export taxes: Duality 4. Evans-Gangbo Performance 5. Optimal Transport Plans 6. Limited importation/exportation

3 Statement of the problem

4 Business A business man (BM) produces some product in some factories in : f + gives the amount of product and its location. There are some consumers of the product in : f gives its distribution. BM wants: to transport all the mass f + to f (to satisfy the consumers) or to the boundary (export).

5 Costs and constrains The BM pays: the transport costs (given by the Euclidean distance) PLUS when a unit of mass is left on a point y : T e (y), the export taxes. BM must satisfy all the demand of the consumers. So that, he has to import product, if necessary, from the exterior, paying: T i (x), import taxes, for each unit of mass that enters from x.

6 Some freedom BM has the freedom to chose to export or import mass provided he transports all the mass in f + and covers all the mass of f.

7 MAIN GOAL The main goal here is to minimize the total cost of this operation, that is given by the transport cost PLUS export/import taxes. Given the set of transport plans { A(f +, f ) := µ M + ( ) : π 1 #µ = f + L N, π 2 #µ = f L N }, the goal is to obtain min µ A(f +,f ) x y dµ + T i d(π 1 #µ) + T e d(π 2 #µ) π i : R N R N are the projections: π 1 (x, y) := x, π 2 (x, y) := y. Given a Radon measure γ in X X, π 1 #γ is the marginal proj x (γ), and π 2 #γ is the marginal proj y (γ).

8 Equilibrium of masses The transport must verify the natural equilibrium of masses, nevertheless it is not necessary to impose f + (x) dx = f (y) dy.

9 Equilibrium of masses The transport must verify the natural equilibrium of masses, nevertheless it is not necessary to impose f + (x) dx = f (y) dy. In this transport problem two masses appear on, that are unknowns, the ones that encode the mass exported and the mass imported. Taking into account this masses we must have the equilibrium condition.

10 Conditions on the domains is assumed to be convex: we prevent the fact that the boundary of is crossed when the mass is transported inside.

11 Conditions on the domains is assumed to be convex: we prevent the fact that the boundary of is crossed when the mass is transported inside. On the support of the f ± we assume that supp(f + ) supp(f ) =.

12 Mass Transport Theory

13 Monge problem Given two measures µ, ν M(X ) (say X R N ) satisfying the mass balance condition dµ = dν, Monge s Problem consists in solving x T (x) dµ(x), inf T #µ=ν X X T : X R N are Borel functions pushing forward µ to ν: T #µ = ν, T #µ(b) = µ(t 1 (B)). X A minimizer T is called an optimal transport map of µ to ν. In the case that µ and ν represent the distribution for production and consumption of some commodity, the problem is to decide which producer should supply each consumer minimizing the total transport cost.

14 Relaxed Problem: Monge-Kantorovich Problem In general, the Monge problem is ill-posed. But...

15 Relaxed Problem: Monge-Kantorovich Problem In general, the Monge problem is ill-posed. But... Fix two measures µ, ν satisfying the mass balance condition. Let Π(µ, ν) the set of transport plans γ between µ and ν: proj x (γ) = µ, proj y (γ) = ν. The problem of finding a measure γ Π(µ, ν) which minimizes the cost functional K(γ) := x y dγ(x, y), in the set Π(µ, ν), is well-posed. X X A minimizer γ is called an optimal transport plan between µ and ν. Linearity makes the Monge-Kantorovich problem simpler than Monge original problem: a continuity-compactness argument guarantees the existence of an optimal transport plan.

16 Dual Problem: Kantorovich Duality Linear minimization problems admit a dual formulation: For (ϕ, ψ) L 1 (dµ) L 1 (dν), define J(ϕ, ψ) := ϕ dµ + ψ dν, X X and let Φ be the set of all measurable functions (ϕ, ψ) L 1 (dµ) L 1 (dν) satisfying Then ϕ(x) + ψ(y) x y for µ ν almost all (x, y) X X. inf K(γ) = sup J(ϕ, ψ). γ Π(µ,ν) (ϕ,ψ) Φ

17 And... Kantorovich-Rubinstein Theorem Theorem (Kantorovich-Rubinstein) where min K(γ) = γ Π(µ,ν) sup u K 1(X ) X u d(µ ν), K 1 (X ) := {u : X R : u(x) u(y) x y x, y X } is the set of 1-Lipschitz functions in X. The maximizers u of the right hand side are called Kantorovich (transport) potentials.

18 Evans and Gangbo Approach For µ = f + L N and ν = f L N, f + and f adequate Lebesgue integrable functions, Evans and Gangbo find a Kantorovich potential as a limit, as p, of solutions to { p u p = f + f in B(0, R), u p = 0 on B(0, R). Theorem u p u K 1 () as p. u (x)(f + (x) f (x)) dx = u(x)(f + (x) f (x)) dx. max u K 1()

19 Evans and Gangbo Approach For µ = f + L N and ν = f L N, f + and f adequate Lebesgue integrable functions, Evans and Gangbo find a Kantorovich potential as a limit, as p, of solutions to { p u p = f + f in B(0, R), u p = 0 on B(0, R). Theorem u p u K 1 () as p. u (x)(f + (x) f (x)) dx = u(x)(f + (x) f (x)) dx. max u K 1() Characterization of the Kantorovich potential: There exists 0 a L () such that f + f = div(a u ) in D (). Furthermore u = 1 a.e. in the set {a > 0}.

20 Evans and Gangbo Approach Evans and Gangbo used this PDE to find a proof of the existence of an optimal transport map for the classical Monge problem, different to the first one given by Sudakov in 1979 by means of probability methods.

21 The mass transport problem with import/export taxes First approach: Duality

22 A Clever Fellow

23 A Clever Fellow A clever fellow proposes the BM to leave him the planing and offers him the following deal:

24 A Clever Fellow A clever fellow proposes the BM to leave him the planing and offers him the following deal: * to pick up the product f + (x), x, he will charge him ϕ(x), * to pick it up at x, T i (x) (paying the taxes by himself), * to leave the product at the consumer s location f (y), y, he will charge him ψ(y), and * for leaving it at y, T e (y) (paying again the taxes himself).

25 A Clever Fellow A clever fellow proposes the BM to leave him the planing and offers him the following deal: * to pick up the product f + (x), x, he will charge him ϕ(x), * to pick it up at x, T i (x) (paying the taxes by himself), * to leave the product at the consumer s location f (y), y, he will charge him ψ(y), and * for leaving it at y, T e (y) (paying again the taxes himself). * He will do all that job in such a way that: ϕ(x) + ψ(y) x y T i ϕ and T e ψ on. (assuming negative payments if necessary).

26 A Clever Fellow A clever fellow proposes the BM to leave him the planing and offers him the following deal: * to pick up the product f + (x), x, he will charge him ϕ(x), * to pick it up at x, T i (x) (paying the taxes by himself), * to leave the product at the consumer s location f (y), y, he will charge him ψ(y), and * for leaving it at y, T e (y) (paying again the taxes himself). * He will do all that job in such a way that: ϕ(x) + ψ(y) x y T i ϕ and T e ψ on. (assuming negative payments if necessary). Observe that we need to impose the following condition: T i (x) T e (y) x y, x, y, (there is not benefit of importing mass from x and exporting it to y). For x = y : T i (x) + T e (x) 0 means that in the same point the sum of exportation and importation taxes is non negative.

27 Clever Fellow s Goal Consider J(ϕ, ψ) := ϕ(x)f + (x) dx + ψ(y)f (y) dy, defined for continuous functions, and let { (ϕ, ψ) C() C() : ϕ(x) + ψ(y) x y, B(T i, T e ) := T i ϕ, T e ψ on The aim of the fellow that helps the business man is to obtain sup {J(ϕ, ψ) : (ϕ, ψ) B(T i, T e )}. }.

28 Clever Fellow s Goal Consider J(ϕ, ψ) := ϕ(x)f + (x) dx + ψ(y)f (y) dy, defined for continuous functions, and let { (ϕ, ψ) C() C() : ϕ(x) + ψ(y) x y, B(T i, T e ) := T i ϕ, T e ψ on The aim of the fellow that helps the business man is to obtain sup {J(ϕ, ψ) : (ϕ, ψ) B(T i, T e )}. Now, given (ϕ, ψ) B(T i, T e ) and µ A(f +, f ): J(ϕ, ψ) = ϕ(x)f + (x) dx + ψ(y)f (y) dy = ϕ(x)dπ 1 #µ ϕdπ 1 #µ + ψ(y)dπ 2 #µ ψdπ 2 #µ }.

29 Clever Fellow s Goal Consider J(ϕ, ψ) := ϕ(x)f + (x) dx + ψ(y)f (y) dy, defined for continuous functions, and let { (ϕ, ψ) C() C() : ϕ(x) + ψ(y) x y, B(T i, T e ) := T i ϕ, T e ψ on The aim of the fellow that helps the business man is to obtain sup {J(ϕ, ψ) : (ϕ, ψ) B(T i, T e )}. Now, given (ϕ, ψ) B(T i, T e ) and µ A(f +, f ): J(ϕ, ψ) = ϕ(x)f + (x) dx + ψ(y)f (y) dy = ϕ(x)dπ 1 #µ ϕdπ 1 #µ + ψ(y)dπ 2 #µ ψdπ 2 #µ x y dµ + T i d(π 1 #µ) + T e d(π 2 #µ). }.

30 Acceptance of the deal Therefore, sup (ϕ,ψ) B(T i,t e) inf µ A(f +,f ) J(ϕ, ψ) x y dµ + T i d(π 1 #µ) + T e d(π 2 #µ).

31 Acceptance of the deal Therefore, sup (ϕ,ψ) B(T i,t e) inf µ A(f +,f ) J(ϕ, ψ) x y dµ + T i d(π 1 #µ) + T e d(π 2 #µ). This inequality will imply that the business man accept the offer.

32 NO GAP But, in fact, there is no gap between both costs: Theorem Assume that T i and T e satisfy T i (x) T e (y) < x y x, y. Then, sup J(ϕ, ψ) (ϕ,ψ) B(T i,t e) = min µ A(f +,f ) x y dµ + T i dπ 1 #µ + T e dπ 2 #µ. Proof based on Fenchel-Rocafellar s duality Theorem.

33 UNKNOWNS There is an important difference between the problem we are studying and the classical transport problem: there are masses, the ones that appear on the boundary, that are unknown variables.

34 UNKNOWNS There is an important difference between the problem we are studying and the classical transport problem: there are masses, the ones that appear on the boundary, that are unknown variables. The above result is an abstract result and does not give explicitly the measures involved in the problem. But...

35 Evans-Gangbo Performance Kantorovich potentials and optimal export/import masses

36 Let f L () and N < p < +. Let g i C( ), g 1 g 2 on. Set Wg 1,p 1,g 2 () = {u W 1,p () : g 1 u g 2 on } and consider the energy functional u(x) p Ψ p (u) := dx f (x)u(x) dx. p

37 Let f L () and N < p < +. Let g i C( ), g 1 g 2 on. Set Wg 1,p 1,g 2 () = {u W 1,p () : g 1 u g 2 on } and consider the energy functional u(x) p Ψ p (u) := dx f (x)u(x) dx. p Wg 1,p 1,g 2 () is a closed convex subset of W 1,p () and Ψ p is convex, l.s.c. and coercive. Then min Ψ p (u) u Wg 1,p 1,g 2 () has a minimizer u p in W 1,p g 1,g 2 (), which is a least energy solution of the obstacle problem { p u = f in, g 1 u g 2 on.

38 Theorem Assume g 1, g 2 satisfy g 1 (x) g 2 (y) x y x, y. Then, up to subsequence, u p u uniformly, and u is a maximizer of the variational problem { } max w(x)f (x) dx : w Wg 1, 1,g 2 (), w L () 1.

39 Theorem Assume g 1, g 2 satisfy g 1 (x) g 2 (y) x y x, y. Then, up to subsequence, u p u uniformly, and u is a maximizer of the variational problem { } max w(x)f (x) dx : w Wg 1, 1,g 2 (), w L () 1. A natural question: Let f + and f be the positive and negative part of f. Can u be interpreted as a kind of Kantorovich potential for some transport problem involving f + and f?

40 DUALITY Theorem If then g 1 (x) g 2 (x) x y x, y, u (x)(f + (x) f (x))dx = sup J(ϕ, ψ) (ϕ,ψ) B( g 1,g 2) = min µ A(f +,f ) x y dµ g 1 dπ 1 #µ + where u is the maximizer given in the above Theorem. g 2 dπ 2 #µ,

41 DUALITY Theorem If then g 1 (x) g 2 (x) x y x, y, u (x)(f + (x) f (x))dx = sup J(ϕ, ψ) (ϕ,ψ) B( g 1,g 2) = min µ A(f +,f ) x y dµ g 1 dπ 1 #µ + where u is the maximizer given in the above Theorem. g 2 dπ 2 #µ, Observe that setting T i = g 1 and T e = g 2 we are dealing with our transport problem.

42 Key result Theorem Assume g 1 (x) g 2 (y) < x y x, y. Let u p, u the functions obtained above. Then, up to a subsequence, X p := Du p p 2 Du p X weakly* in the sense of measures, div(x ) = f in the sense of distributions in, Moreover, the distributions X p η defined as X p η, ϕ := X p ϕ f ϕ for ϕ C0 (R N ), are Radon measures supported on, that, up to a subsequence, X p η V weakly* in the sense of measures; and...

43 Key result u is a Kantorovich potential for the classical transport problem of the measure ˆf + = f + L N + V + to ˆf = f L N + V. Moreover, u = g 1 on supp(v + ), u = g 2 on supp(v ).

44 Key result This result provides a proof for the previous duality Theorem: u d(ˆf + ˆf ) = min x y dν = x y dν 0, ν Π(ˆf +,ˆf ) for some ν 0 Π(ˆf +, ˆf ). Then, since π 1 #ν 0 = V + and π 2 #ν 0 = V, u (f + f ) = x y dν 0 g 1 dπ 1 #ν 0 + g 2 dπ 2 #ν 0.

45 Moreover, we have that V + and V are import and export masses in our original problem.

46 Proof Let u p be a minimizer of Ψ p (u) := u(x) p p dx f (x)u(x) dx. Then, { p u p = f in, g 1 u p g 2 on.

47 Proof Let u p be a minimizer of Ψ p (u) := u(x) p p dx f (x)u(x) dx. Then, { p u p = f in, g 1 u p g 2 on. Therefore for X p := Du p p 2 Du p we have that div(x p ) = f in the sense of distributions, and that the distribution X p η, defined as X p η, ϕ := X p ϕ f ϕ for ϕ C0 (R N ), is supported on.

48 We proof that, in fact, X p η is the sum of a nonnegative and a nonpositive distribution, so it is a measure, and moreover that supp((x p η) + ) {x : u p (x) = g 1 (x)}, and supp((x p η) ) {x : u p (x) = g 2 (x)}. In addition, we have that X p ϕ = f ϕ + ϕ d (X p η) ϕ W 1,p ().

49 Boundedness of u p, X p and X p η By using adequate test functions we get that, for p N + 1, u p p C, C constant not depending on p. d(x p η) ± C,

50 Boundedness of u p, X p and X p η By using adequate test functions we get that, for p N + 1, u p p C, d(x p η) ± C, C constant not depending on p. Therefore, there exists a sequence p i such that u pi u uniformly in, with u 1, and X pi X weakly as measures in, X pi η V weakly as measures on.

51 Then That is, formally, ϕ dx = f ϕ dx + ϕdv div(x ) = f in ϕ C 1 (). X η = V on.

52 Then That is, formally, ϕ dx = f ϕ dx + ϕdv div(x ) = f X η = V in on. ϕ C 1 (). But also we prove that fu + u dv = d X. And this allows to prove that for any 1 Lipschitz continuous function w, u f dx + u dv wf dx + wdv.

53 Giving ϕ C 1 () with ϕ 1, u f dx + u dv = = ϕ dx = ϕf dx + d X ϕdv. X ϕ d X X Then, by approximation, given a Lipschitz continuous function w with w 1, we obtain u f dx + u dv wf dx + wdv.

54 Conclusion Therefore u is a Kantorovich potential for the classical transport problem associated to the measures f + dl N + V + and f dl N + V (the mass of both measures is the same).

55 Conclusion Therefore u is a Kantorovich potential for the classical transport problem associated to the measures f + dl N + V + and f dl N + V (the mass of both measures is the same). Moreover, V + and V are import and export masses in our original problem.

56 Optimal Transport Plans

57 Detailing the transport Once export/import masses on the boundary are fixed, V + and V, take an optimal transport plan µ for f + L N + V + and f L N + V, (remark that necessarily µ( ) = 0). The part of f + exported is f + = π 1 #(µ ), and the part of f imported is f = π 2 #(µ ).

58 An optimal transport plan via transport maps. Kantorovich potential. t 2 : supp( f + ) an optimal map pushing f + forward V +. t 1 : supp( f ) an optimal map pushing f forward V. t 0 : supp(f + ) an optimal map pushing f + f + forward to f f.

59 An optimal transport plan via transport maps. Kantorovich potential. t 2 : supp( f + ) an optimal map pushing f + forward V +. t 1 : supp( f ) an optimal map pushing f forward V. t 0 : supp(f + ) an optimal map pushing f + f + forward to f f. Then, µ (x, y) = f + (x)δ y=t2(x) + (f + (x) f + (x))δ y=t0(x) + f (y)δ x=t1(y) is an optimal transport plan for our problem. u is a Kantorovich potential for each of these transports problems.

60 u (x) = min y (g 2(y) + x y ) for a.e. x supp( f + ), u (x) = max y (g 1(y) x y ) for a.e. x supp( f ).

61 T i = 0, T e = 1 2 y x

62 T i = 0, T e = 1 2 y x Importing mass from 1 is free of taxes, nevertheless, doing this would imply to export more mass to 0 and export taxes make this expensive.

63 T i = 0, T e = 1 4 y x Decreasing the export taxes, more mass is exported to 0 and some mass is imported from 1.

64 T i = 0, T e(0) = 3 4, Te(1) = 0 y x Increasing export taxes in 0, all the other null, implies that the optimal transport plan exports to 1.

65 T i = 0, T e = 1 2 y x

66 T i = 0, T e = 1 2 y x Observe that this transport problem would have coincided with the classical one if we had put T e = 1.

67 Limited importation/exportation

68 More realistic We have been using as an infinite reserve/repository.

69 More realistic We have been using as an infinite reserve/repository. We can impose the restriction of not exceeding the punctual quantity of M e (x) when we export some mass through x and also a punctual limitation of M i (x) for importing from x. We will assume M i, M e L ( ) with M i, M e 0 on.

70 More realistic We have been using as an infinite reserve/repository. We can impose the restriction of not exceeding the punctual quantity of M e (x) when we export some mass through x and also a punctual limitation of M i (x) for importing from x. We will assume M i, M e L ( ) with M i, M e 0 on. A natural constraint must be verified: M e (f + f ) that is, the interplay with the boundary is possible. Hence the mass transport problem is possible, and the problem becomes, as before, to minimize the cost. M i,

71 Especial situations 1. When moreover T i = T e = 0: limited importation/exportation but without taxes. 2. When M i (x) = 0 or M e (x) = 0 on certain zones of the boundary: it is not allowed importation or exportation in those zones. 2. M i (x) = M e (x) = 0 on the whole boundary ( f + = f ) we recover the classical Monge-Kantorovich mass transport problem.

72 New goal Given now A l (f +, f ) := µ M + ( ) : π 1 #µ = f + L N, π 2 #µ = f L N, π 1 #µ M i, π 2 #µ M e, we want to obtain min µ A l (f +,f ) x y dµ + T i d(π 1 #µ) + T e d(π 2 #µ). Conditions on T i and T e : T i + T e 0 on.

73 The Clever Fellow again New deal: * to pick up the product f + (x), x, he will charge him ϕ(x), * to pick it up at x, T i (x) (paying the taxes by himself) * to leave the product at the consumer s location f (y), y, he will charge him ψ(y), and * for leaving it at y, T e (y) (paying again the taxes himself).

74 The Clever Fellow again New deal: * to pick up the product f + (x), x, he will charge him ϕ(x), * to pick it up at x, T i (x) (paying the taxes by himself) * to leave the product at the consumer s location f (y), y, he will charge him ψ(y), and * for leaving it at y, T e (y) (paying again the taxes himself). * he will do all this in such a way that: ϕ(x) + ψ(y) x y T i ϕ and T e ψ on,

75 The Clever Fellow again New deal: * to pick up the product f + (x), x, he will charge him ϕ(x), * to pick it up at x, T i (x) (paying the taxes by himself) * to leave the product at the consumer s location f (y), y, he will charge him ψ(y), and * for leaving it at y, T e (y) (paying again the taxes himself). * he will do all this in such a way that: ϕ(x) + ψ(y) x y T i ϕ and T e ψ on, (now:) * he will pay a compensation when ϕ < T i or ψ < T e of: M i ( ϕ T i ) + + M e ( ψ T e ) +.

76 The aim of the fellow now is to obtain ϕf + + ψf M i ( ϕ T i ) + sup ϕ(x)+ψ(y) x y M e ( ψ T e ) +.

77 The aim of the fellow now is to obtain ϕf + + ψf M i ( ϕ T i ) + sup ϕ(x)+ψ(y) x y M e ( ψ T e ) +. And again There is no gap between both costs.

78 New energy functional The energy functional to be considered now is u(x) p Ψ l (u) := dx f (x)u(x) dx + j(x, u(x)), p M i (x)( T i (x) r) if r < T i (x), j(x, r) = 0 if T i (x) r T e (x), M e (x)(r T e (x)) if r > T e (x). For the minimizers of we prove the following result: min Ψ l(u), u W 1,p ()

79 Theorem Up to subsequence, u p u uniformly as p.

80 Theorem Up to subsequence, u p u uniformly as p. There exist X p η L ( ), X p η j(x, u p ) a.e. x such that X p η ϕ = Du p p 2 Du p ϕ f ϕ for all ϕ W 1,p (), so u p are solutions of the nonlinear boundary problem { p u = f in, u p 2 u η + j(, u( )) 0 on.

81 Theorem Up to subsequence, u p u uniformly as p. There exist X p η L ( ), X p η j(x, u p ) a.e. x such that X p η ϕ = Du p p 2 Du p ϕ f ϕ for all ϕ W 1,p (), so u p are solutions of the nonlinear boundary problem { p u = f in, u p 2 u η + j(, u( )) 0 on. X p η V weakly in L ( ) with V + M i, V M e.

82 Theorem sup w(x)f (x) dx j(x, w(x)) w W 1, (), w L () 1 = u (x)(f + (x) f (x))dx j(x, u (x)) = sup ϕ(x)+ψ(y) x y ϕf + + ψf M i ( ϕ T i ) + M e ( ψ T e ) + = min x y dµ + T i dπ 1 #µ + T e dπ 2 #µ. µ A l (f +,f )

83 u is a Kantorovich potential for the classical transport problem for the measures f + L N + V + dh N 1 and f L N + V dh N 1. V + M i and V M e are the import and export masses.

84 T i = 0, T e = 1 2, M i = M e = y x Cost: 56/16 2.

85 T i = 0, T e = 1 2, M i = M e = 1 16 y x Cost: 58/16 2.

86 Some references L. V. Kantorovich. On the tranfer of masses, Dokl. Nauk. SSSR 37 (1942), V. N. Sudakov. Geometric problems in the theory of infinite-dimensional probability distributions, Proc. Stekelov Inst. Math., 141 (1979), L. C. Evans and W. Gangbo. Differential equations methods for the Monge-Kantorovich mass transfer problem. Mem. Amer. Math. Soc., L. C. Evans. Partial differential equations and Monge-Kantorovich mass transfer. Current developments in mathematics, L. Ambrosio. Lecture notes on optimal transport problems, L. Ambrosio and A. Pratelli. Existence and stability results in the L 1 theory of optimal transportation. Lecture Notes in Math., C. Villani. Topics in Optimal Transportation. Graduate Studies in Mathematics, C. Villani. Optimal Transport, Old and New. Fundamental Principles of Mathematical Sciences, 2009.

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