1. Solving For User Equilibrium Chapter 5
Transportation Systems Analysis
Solving For User Equilibrium Chapter 5
Meeghat Habibian

2. CONVEX COMBINATIONS METHOD
This method is especially suitable for solving the equivalent UE program since the direction- finding step can be executed relatively efficiently. This step involves the solution of a linear program, which, in the case of the UE program, has a special structure that simplifies its solution.

3. The UE problem
Applying the convex combinations algorithm to the minimization of the UE program requires, at every iteration, a solution of the linear program (LP)

4. The LP objective function at the nth iteration:
Where
The program in Eqs. [5.3] is known as the all-or-nothing program.

5. The gradient of z(x) with respect to the link flows at the nth iteration is the link travel-time vector, since ∂z(xn )/ ∂ xa =tan
ya :the auxiliary variable representing the flow on link a
grsk : the auxiliary flow variable for path k connecting O-D pair r-s

6. To see that the solution of program [5
To see that the solution of program [5.3] does not involve more than an all-or-nothing assignment, the linearization step of the convex combinations method can be derived by taking the gradient of Objective function [5.la] with respect to path flows (instead of link flow, as in [5.2]).

7. The linearized program then becomes:
Ckrs n : travel time on path k connecting r and s at the nth iteration

8. This program can be decomposed by O-D pair since the path travel times are fixed. The resulting sub problem for pair r-s is given by
Finding the min path ,m,

9. In case two or more paths are tied for the minimum, any one of them can be chosen for flow assignment. Once the path flows {gkrsn} are found, the link flows can be calculated by using the incidence relationships, that is:

10. This solution defines the descent direction
dn = yn – xn
The initial solution
The initial solution can be usually determined by applying an all-or-nothing network loading procedure to an empty network.

11. The line search for the optimal move size can be performed with any of the interval reduction methods, but the bisection method (Bolzano search) may be particularly applicable. The reason is that the derivative of the Objective function z[xn+ α(yn - xn)] with respect to a is given by which can be easily calculated for any value of α.

12. Stopping criterion for solving the UE program
The convergence criterion should be based on the
relevant figures of merit, which in this case consist of the flows and the travel times.
ursn :minimum path travel time between O-D pair r-s at the nth iteration
Other convergence criteria can be used as well.

13. The algorithm Step 0: Initialization. Perform all-or-nothing
assignment based on t0a= ta (0), This yields {xa1}. Set counter n = 1.
Step 1: Update. Set tna = ta(xan),

14. Step 2: Direction finding
Step 2: Direction finding. Perform all-or-nothing assignment based on {tna}. This yields a set of (auxiliary) flows {yan}
Step 3: Line search. Find αn that solves
Step 4: Move. Set xan+1= xan + αn (yan - xan ),

15. Step 5: Convergence test
Step 5: Convergence test. If a convergence criterion is met, stop (the current solution, {xan+1 }, is the set of equilibrium link flows); otherwise, set n := n + 1 and go to step 1.

16. Example: the network with three links and one O-D pair
Figure5.1

18. Comparison of this algorithm with heuristic methods
At this point, the travel times generated by the convex combinations algorithm on the three paths are t4 =(26.1, 26.3, 25.3) This situation is much closer to equilibrium than the other methods. Incremental assignment ,t* = (69.0, 27.3, 25.0)

19. As Table 5.4 shows, flow is taken away from congested paths and assigned to less congested paths, at each iteration of the convex combinations method.
The objective function value decrease in each successive iteration. The marginal contribution of each successive iteration to the reduction in the objective function value is decreasing.

20. The number of iterations required for convergence is significantly affected by the congestion level on the network. In relatively uncongested networks, a single iteration may suffice since the link flows may be in the range where the performance functions are almost flat.
This effect is demonstrated in Figure 5.2, which depicts the convergence pattern of the convex combinations method for a medium-sized network, for three congestion levels.

22. Congestion is measured in terms by:
In Figure 5.2Convergence rate of the convex combinations algorithm for various congestion levels.
Congestion is measured in terms by:
Convergence is measured by the value of the objective function, normalized between the initial iteration and the 30th one.
Convergence is measured at iteration (n) by:

23. Thanks for your attention