Optimum and equilibrium in a transport problem with queue - PowerPoint PPT Presentation

Optimum and equilibrium in a transport problem with queue penalization effect. G. Crippa, (University of Parma (Italy)) C. Jimenez, (Université de Bretagne Occidentale) A. Pratelli, (University of Pavia (Italy).) C. Jimenez, UBO JFCO 2008

Presentation of the Problem City: Ω ⊂ R d bounded open set, k post-offices : x 1 , ..., x k ∈ Ω fixed, Population density: f dx a probability, Unknown: partition ( A i ) i = 1 ,... k of Ω : every person linving on A i goes to x i . Time lost by a citizen living at x ∈ A i : � number of persons going to x i : c i = A i f ( x ) dx Time = Displacement + queue | x − x i | p = + h i ( c i ) p ≥ 1 . k � ( | x − x i | p + h i ( c i )) f ( x ) dx . � Total cost = A i i = 1 C. Jimenez, UBO JFCO 2008

Associated optimization problem � �� � �� � � | x − x i | p f ( x ) dx + � k inf ( A i ) i f ( x ) dx h i f ( x ) dx i = 1 A i A i A i � ( A i ) i = 1 ,... k partition of Ω k � � � | x − x i | p f ( x ) dx : � = inf ( c i ) i inf ( A i ) i A i i = 1 partition of Ω � � A i f ( x ) dx = c i , c i ≥ 0 � + � i c i h i ( c i ) : � i c i = 1 w p p ( fdx , � k i = 1 c i δ x i ) + � k � i = 1 c i h i ( c i ) : c i ≥ 0 , � � = inf ( c i ) i i c i = 1 W p ( fdx , � k i = 1 c i δ x i ) is the p -Wasserstein distance. C. Jimenez, UBO JFCO 2008

Optimal Transportation Wasserstein distance from fdx to � k i = 1 c i δ x i k � � | x − Tx | p f ( x ) dx W p ( fdx , c i δ x i ) = inf T Ω i = 1 T ( x ) = x i ∀ x ∈ A i where ( A i ) i = 1 ,... k is a partition Ω such that: � A i f ( x ) dx = c i . Existence of a transport map It exists an optimal transport map T for W p . p = 2 Brenier (87). p > 1 Rüschendorf (95), Gangbo, McCann (96). p = 1 Sudakov (79), Gangbo, McCann (96), Evans, Gangbo (99), Caffarelli, Feldmann, McCann (02), Ambrosio, Pratelli (03)... C. Jimenez, UBO JFCO 2008

Optimality transportation Kantorovich duality W p ( fdx , � k i = 1 c i δ x i ) k � � � u ∈ L 1 (Ω) , = sup u , ( α i ) i u ( x ) f ( x ) dx + c i α i : Ω i = 1 � u ( x ) − α i ≤ | x − x i | p a.e. x ∈ Ω C. Jimenez, UBO JFCO 2008

Optimal transportation Primal-Dual Optimality conditions (Bouchitté)  T = � i x i 1 A i  u ( x ) = inf i | x − x i | p − α i �  ( u , ( α i ) i , T )  ⇒ i ( | x − x i | p − α i ) 1 A i = � optimal  A i = { x ∈ Ω : | x − x i | p − α i < | x − x i | p − α j }   � T = � u ( x ) = inf i | x − x i | p − α i � i x i 1 A i and ⇒ i ( | x − x i | p − α i ) 1 A i = � u , ( α i ) i optimal. Consequence: the optimal partition is always unique. C. Jimenez, UBO JFCO 2008

The optimization problem inf ( c i ) i { W p p ( fdx , � k i = 1 c i δ x i ) + � k i = 1 c i h i ( c i ) : c i ≥ 0 , � i c i = 1 } k � W p � � p ( fdx , c i δ x i ) = inf { | x − x i | f ( x ) dx } ( A i ) i A i i = 1 i � under the constraints: ( A i ) i is a partition of Ω , c i = f ( x ) dx . A i Existence of an optimum If t �→ th i ( t ) is l.s.c. for any i = 1 , .. k , it exists an optimal partition of Ω . Moreover if t �→ th i ( t ) is strictly convex, the optimal partition is unique. C. Jimenez, UBO JFCO 2008

The optimization problem Necessary and Sufficient Optimality Condition We assume h i is regular and t �→ th i ( t ) is convex for any i = 1 , .. k , then a partition ( A i ) i is optimal iff (up to negligible sets): | x − x i | p + h i ( c i ) + c i h ′ A i = { x ∈ Ω :  i ( c i ) < | x − x j | p + h j ( c j ) + c j h ′  j ( c j ) } � c i = A i f ( x ) dx .  Moreover, there is only one partition which satisfies this condition. C. Jimenez, UBO JFCO 2008

Example Let Ω = [ 0 , 1 ] , f = 1, x 1 = 0, x 2 = 1, p = 1 and: � 0 for 0 ≤ s ≤ 0 . 999 h 1 ( s ) = 100 , and h 2 ( s ) = 1 for 0 . 999 < s ≤ 1 . Optimum: A 1 = [ 0 , 0 . 001 [ , A 2 =] 0 . 001 , 1 ] ... Costumers in A 1 may not be happy! A costumer living at x ∈ A i will be happy if: | x − x i | p + h i ( c i ) = inf j {| x − x j | p + h i ( c j ) } . In the example: A 1 = ∅ , A 2 = [ 0 , 1 ] . C. Jimenez, UBO JFCO 2008

The equilibrium problem Nash Equilibrium A partition ( A i ) i = 1 ,.. k is a Nash equilibrium if: � A i = { x ∈ Ω : | x − x i | p + h i ( c i ) < | x − x j | p + h j ( c j ) } � c i = A i f ( x ) dx . Existence Assume h i is continuous for any i = 1 , .. k and g i is such that tg ′ i ( t ) + g i ( t ) = h i ( t ) . Then, the minimizer of the following problem is an equilibrium: �� � �� � � | x − x i | p + g i inf f ( x ) dx f ( x ) dx . ( A i ) i A i A i i partition of Ω If, in addition, h i is non-decreasing, the equilibrium is unique. C. Jimenez, UBO JFCO 2008

Example Travellers game Traveller 1 Traveller 2 2 ≤ t 1 ≤ 100 2 ≤ t 2 ≤ 100 t 1 ∈ N t 2 ∈ N t 1 = t 2 t 1 t 2 t 1 < t 2 t 1 + 2 t 1 − 2 t 1 > t 2 t 2 − 2 t 2 + 2 Equilibrium: t 1 = t 2 = 2. C. Jimenez, UBO JFCO 2008

Pareto Optimum If ( A i ) i is an equilibrium, is it possible to find another partition ( B i ) i such that every citizen spends equal or less time ? Individual cost : k � �� �� | x − x i | p + h i � C ( x , ( B i ) i ) = f ( x ) dx 1 B i ( x ) . B i i = 1 Pareto Optimum A partition ( A i ) i = 1 ,.. k is a Pareto optimum if there exists NO other partition ( B i ) i of Ω such that: C ( x , ( B i ) i ) < C ( x , ( A i ) i ) a.e. x ∈ Ω . Proposition Assume h i is strictly increasing. Then every equilibrium is a Pareto optimum. C. Jimenez, UBO JFCO 2008

Sketch of proof Important remark ( A i ) is an equilibrium ⇔ � | x − x i | p + h i �� �� C ( x , ( A i ) i ) = inf i = 1 ,... k A i f ( x ) dx . Remember the optimality condition for optimal transportation (slide 5) C. Jimenez, UBO JFCO 2008

Sketch of proof Link between equilibrium and optimal transportation If ( A i ) i is an equilibrium then: �� � T = � i x i 1 A i is optimal for W p ( f , � A i f ( x ) dx δ x i ) , i �� � u ( x ) = C ( x , ( A i ) i ) and ( α i ) i = − h i A i f ( x ) dx are optimal for the dual formulation of W p . If ( A i ) i is a partition such that u ( x ) = C ( x , ( A i ) i ) and �� � ( α i ) i = − h i A i f ( x ) dx are optimal for the dual formulation of W p , then: ( A i ) i is an equilibrium �� � T = � i x i 1 A i is optimal for W p ( f , � A i f ( x ) dx δ x i ) . i C. Jimenez, UBO JFCO 2008

Sketch of proof Assume h i ր , ( A i ) i is an equilibrium, ( B i ) i is such that C ( x , ( B i ) i ) ≤ C ( x , ( A i ) i ) a.e. � � Let us show: A i f ( x ) dx = B i f ( x ) dx for every i . � � Assume ∃ j such that A j f ( x ) dx < B j f ( x ) dx . Then: � | x − x j | p + h j ( ∀ x ∈ B j f ( x ) dx ) B j = C ( x , ( B i ) i ) ≤ C ( x , ( A i ) i ) � i {| x − x i | p + h i ( = min f ( x ) dx ) } A i � | x − x j | p + h j ( ≤ f ( x ) dx ) . A j This is impossible because h j is strictly increasing. C. Jimenez, UBO JFCO 2008

Sketch of proof As ( A i ) i is an equilibrium, T = � i x i 1 A i is an optimal �� � transport map for W p ( f , � A i f ( x ) dx δ x i ) , by i consequence: � � | x − x i | p f ( x ) dx ≤ | x − x i | p f ( x ) dx . � � A i B i i i But as C ( x , ( B i ) i ) ≤ C ( x , ( A i ) i ) and � � A i f ( x ) dx = B i f ( x ) dx : � � | x − x i | p f ( x ) dx ≥ | x − x i | p f ( x ) dx . � � A i B i i i By uniqueness of the optimal transport map, ( A i ) i = ( B i ) i up to negligible sets. C. Jimenez, UBO JFCO 2008

Dynamic behaviour for 2 points For simplicity, assume: Ω = [ 0 , 1 ] , x 1 = 0 , x 2 = 1 , f = 1 , p = 1 . At every day n , we set t n such that: A 1 = [ 0 , t n [ , A 2 =] t n , 1 ] . Day 1: the citizen has no information on the queue: t 1 = 1 / 2, End of day: he knows h 1 ( t 1 ) , h 2 ( 1 − t 1 ) . Day n : he knows h 1 ( t n − 1 ) , h 2 ( 1 − t n − 1 ) , t n + 1 + h 1 ( t n ) = ( 1 − t n + 1 ) + h 2 ( 1 − t n ) t n + 1 := h 2 ( 1 − t n ) − h 1 ( t n ) + 1 . 2 t = h 2 ( 1 − ¯ t ) − h 1 (¯ Equilibrium: ¯ t such that: ¯ t )+ 1 . 2 C. Jimenez, UBO JFCO 2008

Dynamic behaviour for 2 points Convergence Let G ( t ) := h 2 ( 1 − t ) − h 1 ( t )+ 1 . If G is a contraction mapping, then 2 t n → ¯ t . C. Jimenez, UBO JFCO 2008

Dynamic with memory Assume the citizen "remembers" the K last days. The map G is the same as before but we set: � K i = 1 G ( t n − K + i ) t n + 1 = . K Convergence Assume G is L -lipschitz with L < K , then t n → t . C. Jimenez, UBO JFCO 2008

Sketch of proof for K = 2 Assume that for every k ≤ n : � ( i ) | G ( t k ) − G (¯ t ) | ≤ α | G ( t k ) − G (¯ t ) + G ( t k − 1 ) − G (¯ ( ii ) t ) | ≤ α Recall t n + 1 = G ( t n )+ G ( t n − 1 ) . 2 | G ( t n + 1 ) − G (¯ L | t n + 1 − ¯ t ) | ≤ t | Lipschitz Property L | t n + 1 − G (¯ ¯ t ) | = t is a fixed point G ( t n ) − G (¯ t ) + G ( t n − 1 ) − G (¯ � � t ) � � = L � � 2 � � L α ≤ by (ii) . 2 C. Jimenez, UBO JFCO 2008