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1 Equilibrio come punto fisso User Equilibrium no user finds convenient to unilaterally change path fixed-point formulations express the circular dependency of flows and costs in terms of path variables supply modelC(F) = Δ T c(Δ F) + G demand modelF(C) = P(C) diag(D T ) F = F(C(F)) in terms of link variables (no G here) arc cost functionc(f) network loading mapf NLM (c) = Δ P(Δ T c) diag(D T ) f = f NLM (c(f)) for deterministic models substitute = with

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2 Dipendenza circolare C c f Δ F P(C) diag(D T ) c(f) Δ T c+G F

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3 Esistenza del punto fisso generic fixed point problem x = (x), x X sufficient conditions for existence (x) is C 0 X is non-empty, compact, convex (X) X (X) (x) X X x X x (X) (x) x (X) (x) X x

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4 Unicità del punto fisso condizioni sufficienti per lunicità (x) monotonica (X) (x) X x x X (X) (x)

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5 Soluzione del punto fisso contrazione converge se e solo se il punto fisso è una contrazione lo user equilibrium non è una contrazione x h+1 = (x h ) metodo delle medie successive (MSA) il nuovo punto è la media di tutti i punti precedenti converge ad un punto fisso y h = (x h ) x h+1 = x h +1/h (y h -x h ) lMSA può essere esteso considerando una media pesata dei punti precedenti

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6 Unicità dellequilibrio if the choice map is monotone non increasing [P(C 1 )-P(C 2 )] T (C 1 -C 2 ) 0 C 1,C 2 the network loading map is such [f NLM (c 1 )-f NLM (c 2 )] T (c 1 -c 2 ) 0 c 1,c 2 if the arc cost function is monotone increasing [c(f 1 )-c(f 2 )] T (f 1 -f 2 ) > 0 f 1 f 2 the UE is unique in terms of link flows assume by contradiction that f 1 and f 2 are both equilibrium flow vectors then f 1 = f NLM (c(f 1 )) and f 2 = f NLM (c(f 2 ))

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7 MSA applicato allequilibrio 0) k = 0, f k+1 = 0 punto iniziale 1) k = k+1 nuova iterazione 2) c k = c(f k ) + k funzione di costo darco 3) y k = f NLM (c k ) mappa di carico della rete 4) f k+1 = f k +1/k (y k -f k ) media dei flussi o passo 5) se ||f k+1 -f k ||/||f k ||< STOP criterio di stop 6) torna al passo 1

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8 Deterministic route choice and UE users travel only on shortest paths the minimum cost between o R and d R is the smallest path cost among those belonging to the set K od Z o d = min{C k : k K od } if C k = Z o d then path k K od is shortest Wardrops principles 1) any used path (F k > 0) is shortest ( C k = Z o d ) 2) any non-shortest path (C k > Z o d ) is unused ( F k = 0) formulation of the deterministic route choice model F k (C k - Z o d ) = 0 k K od, o R, d R C k Z o d k K od, o R, d R F k 0 k Kod, o R, d R k Kod F k = D od o R, d R if C = T c( F), this condition defines a User Equilibrium

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9 Equivalent optimization convex program with sum-integral objective function (Beckmann et. al., 1956) assumption for the existence of a solution c ij non-negative and continuous K od non-empty D od non-negative assumption for its uniqueness in terms of link flows c ij strictly monotone increasing its first order (necessary) conditions coincide with the formulation of the deterministic user equilibrium

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10 Necessary conditions Lagrangian problem min{L(F, Z): F 0} L(F, Z) = ij A 0 k K F k ij k c ij (v) dv + od Z o d (D od - k Kod F k ) Z o d are here the Lagrangian multipliers necessary conditions F k L(F, Z)/ F k = 0 k K od, o R, d R L(F, Z)/ F k 0 k K od, o R, d R L(F, Z)/ Z o d = 0 o R, d R F k 0 k K od, o R, d R since it is L(F, Z)/ F k = ij A c ij ij k - Z o d = C k - Z o d we obtain agian the formulation of the deterministic route choice model

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11 Variational inequalities Beckmann program in terms of path flows the gradient of (F) is C = T c( F) linearize the objective function at a given a point F S F (Y) = (F) + C T (Y-F) + o(1/||Y-F||) a direction Y-F is descent at point F if and only if C T (Y-F) < 0 a point F S F is a local minimum only if C T (Y-F) 0 Y S F this condition for equilibrium is more general than Beckmann program and allows for asymmetric arc cost functions the problem is not unique in terms of path flows

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12 Descent algorithms direction Y-F is descent at point F if and only if C T (Y-F) < 0 to decrease the objective function and maintain feasibility we have to shift path flows lowering the total cost with respect to the current cost pattern let Y S F define a feasible descent direction since S F is convex we can move the current solution along the segment F+ (Y-F) and take a step (0,1] such that the original objective function gets lowered

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13 Step size the objective function is C 1 and convex we then have two alternative approaches minimize along the segment (e.g. using the bisection method) min{ (F+ (Y-F)): (0,1]} (line search) determine the largest step = 0.5 n, with n positive integer such that (F+0.5 n (Y-F))/ < 0 (Armijo-like search) (F+ (Y-F))/ = c( (F+ (Y-F))) T (Y-F) = c(f+ (y-f)) T (y-f) that can be applied to each o-d pair separately to each origin or destination (yielding the Origin-Based algorithms) to the whole network jointly

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14 Frank-Wolfe obtaining Y argmin{C T X: X S F } through an All-Or-Nothing assignment to shortest paths provides a descent direction adopt such a direction for all o-d pairs of the network jointly and apply along it a line search the result is the well known Frank-Wolfe algorithm (or convex combination) it solves convex non-linear optimization problems with linear contraints the procedure 0) h = 0, find a feasible point f h – AoN based on the link costs c 0 = c(0) 1) linearize the objective function in f h and find a solution y h to the linearized problem – AoN based on the link costs c h = c(f h ) 2) perform a line search along the segment from f h to y h finding a new point f h+1 – apply the bisection method 3) if the convergence criteria on the gap function is satisfied then STOP – c h T (f h -y h ) / c h T y h < 4) h = h +1, go back to step 1)

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15 Graphical interpretation feasible set tangent hyper-plan fhfh f h+1 yhyh - (f h ) objective function

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16 Bisection method find the minimum of a function f(x): which is C 1 (continuous with continuous first derivative) strictly convex (the function is above the tangent at any point defined in a closed interval [a, b] ba y = 0.5 (a +b) df(y)/dx f(x) x at each iteration the interval is made a half do until 0.5 (b - a) < y = 0.5 (a + b) if df(y)/dx < 0 then a = y else b = y

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17 Algorithms for traffic assignment Path based Gradient Projection (Jayakrishnan et. al., 1994) Restricted Simplicial Decomposition (Hearn et. al., 1987) Convex programming Frank-Wolfe (LeBlanc, 1973; Nguyen, 1973) PARTAN (florian et. al., 1987) Fixed point methods Method of Successive Averages iGSM (Bierlaire and Crittin, 2006) Bathers method (Bottom and Chabini, 2001) Origin-Based algorithms OBA (Bar-Gera, 2002) TAPAS (Bar-Gera, 2006) Algorithm B (Dial, 2006) Linear Cost Equilibrium (Gentile, 2006)

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