1. 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

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

3. 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

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

5. 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

6. 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 ))

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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)

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

16. 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

17. 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)