Globally convergent line search algorithm with Euler-based step size-determination method for continuous network design problem

Most solution algorithms for the continuous network design problem (CNDP) follow the line search principle. However, they mainly focus on identifying a feasible descent direction in each iteration. No efficient approaches have been proposed to determine a step size in each iteration to enhance the convergence speed because the equilibrium flows in CNDP are implicit functions of the decision variables. To bridge this gap, this study proposes a norm-relaxed method of feasible direction (NRMFD) algorithm integrated with the Euler-based approximation (EBA) method for the CNDP under user equilibrium (UE). At each iteration, after a feasible descent direction is determined by solving a quadratic program, this algorithm computes a feasible step size to reduce the value of the objective function using the EBA method. The main idea of the EBA method is to adaptively generate a sequence of candidate step sizes and estimate the respective UE link flows and objective function. The feasible step size that reduces the objective function the most is accepted in the NRMFD algorithm to achieve a faster convergence speed. The EBA method is computationally efficient as it linearly approximates the UE solution rather than solving it precisely. The NRMFD algorithm integrated with the EBA method is analytically proved to be globally convergent. Numerical examples indicate that the proposed algorithm can efficiently and effectively solve the CNDP in a few iterations.