A Class of Efficient Algorithms for the Bi-Level Demand Adjustment Problems in Congested Traffic Networks

Algorithm 2

Algorithm for solving the quadratic programming model.

	Step 1: Set the initial solution:
	Step 2: Calculate the descent direction:
	For link , the descent direction can be obtained using equation (5b), and for path , the descent direction can be obtained using equation (5a). As a result, the descent direction of the path-based objective function is , and the descent direction of the link-based objective function is .
	Step 3: Calculate the optimal stepsize:
	The optimal stepsize can be determined as follows: . Then, we update and to and .
	Step 4: Check the convergence criterion:
	Examine whether the projection gradient is less than (e.g., ), terminate the algorithm if it is satisfied and go to Step 2 otherwise.
	Note the optimal stepsize along the descent direction for the quadratic programming model in Step 3 can be obtained by solving: