Mathematical programming model and optimization algorithms of graph theory based network design problem

This thesis addresses the optimization of traffic flows in transportation networks by examining Capacitated Traffic Assignment Problems (CTAP) as dynamic systems. Recognizing the limitations of traditional static models in capturing time-dependent variations of traffic flows, which hinder their full adaptability to real-world conditions, this study aims to transform static CTAP into a dynamic system based on Lagrange Neural Networks (LSA). The proposed model seeks to provide a more realistic representation of the temporal behavior of traffic flows while ensuring asymptotic stability through Lyapunov stability analysis and projection-based numerical methods, which are rigorously validated mathematically. Methodologically, the study adapts the LSA framework to CTAP and employs projection operators to maintain the physical consistency of primal-dual variables. This approach guarantees adherence to non-negativity constraints for variables such as path flows \( h_\eta \), origin-destination (O-D) demand Lagrange multipliers \( \lambda_{rs} \), and delay Lagrange multipliers \( \nu_{ij} \). Additionally, a normalization technique is applied to ensure the satisfaction of O-D demand constraints \( \sum_{\eta \in \eta_{rs}} h_\eta = d_{rs} \). The numerical solution of the model is implemented using the fourth-order Runge-Kutta (RK4) method and tested on well-established traffic networks, including Hearn and Sioux Falls. Numerical analyses demonstrate that the proposed dynamic system stably converges to Wardrop equilibrium conditions and delivers efficient solutions under capacity constraints. The results highlight the model's capability to replicate realistic traffic flow dynamics while maintaining rigorous mathematical stability, thereby bridging the gap between theoretical frameworks and practical applicability in transportation network optimization.