Frank-Wolfe Algorithm for Traffic Assignment in Transportation Network Flow

The Frank-Wolfe algorithm is an iterative optimization method designed for solving convex optimization problems, with significant applications in traffic assignment problems within transportation network flows. This algorithm demonstrates particular effectiveness for path flow allocation in large-scale transportation networks where computational efficiency is critical.

The core objective of traffic assignment involves distributing travel demand appropriately across network paths to achieve system equilibria such as User Equilibrium (UE) or System Optimal (SO) states. The Frank-Wolfe algorithm employs an iterative approach that progressively converges toward optimal solutions. Each iteration computes an auxiliary direction through linear programming subproblems, followed by a line search operation to update the current solution vector. In code implementation, this typically involves maintaining arrays for current flows, computing descent directions using shortest path algorithms, and performing step size calculations.

The algorithm's primary advantages in traffic assignment applications include low memory requirements and straightforward implementation architecture. It is especially suitable for Beckmann-type traffic assignment models, where the problem decomposes into sequential linear subproblems - effectively reducing computational complexity. Code implementation typically utilizes sparse matrix operations for network representation and employs efficient shortest-path algorithms (like Dijkstra's method) for direction finding at each iteration.

While the convergence rate may slow near optimal solutions (particularly due to its first-order nature), the algorithm provides rapid solution improvement during early iterations, making it highly valuable for practical large-scale network applications. Subsequent enhanced algorithms like Method of Successive Averages (MSA) often build upon the Frank-Wolfe framework by incorporating improved step size strategies and convergence acceleration techniques.