conjugate direction algorithms

The Traffic assignment problem in the transport networks is formulated as a convex optimization problem using simplifier assumptions. Link based, path based and bush based algorithms have been presented to solve this problem, among which the link based algorithm have found more applications thanks to their low memory requirements. The link based algorithm of Frank-Wolf (FW) is yet amongst the most popular assignment algorithms, because of its simplicity as well as high convergence speed during the initial iterations. However, the low convergence rate of this algorithm near the optimal solution has driven many studies that focus on modifying the FW search direction, resulted in newer link based algorithms. Among them are conjugate direction algorithms which are more effective and simply implementable. These algorithms include PARTAN, conjugate FW (CFW) and bi-conjugate FW (BFW). This paper uses the Chicago Regional and Sioux-Falls test networks to make direct comparisons among these algorithms with respect to the CPU time and iteration number needed to reach various accurate solutions. The results for Chicago show that, compared with the FW algorithm, the three algorithms CFW, PARTAN increase the convergence speed to the relative error of 10-5 (i.e. a stable solution) by about 89, 72 and 63%, respectively, while only the BFW can reach to a 10-6 relative error. Comparing the results from Sioux-Fall with those of Chicago demonstrate that the performance of the conjugate directions Algorithms improves as network size decreases.