On the Convergence of an Efficient and Robust Dynamic Neural Network Concept with Application to Solving Traveling Salesman Problems

Abstract

In our previous contributions [20, 21, 22], we have clearly demonstrated that the dynamic neural network concept (DNN-concept) for solving shortest path problems (SPP) and traveling salesman problems (TSP) outperforms the best heuristic methods proposed by the literature. However, in our numerous contributions and also according to the literature, the effects of the step sizes of both “decision neurons” and “multiplier neurons” on the convergence properties of the “DNN-concept” are still not investigated. The aim of our contribution is to enrich the literature by investigating, for the first time, the convergence properties of the DNN-concept for solving traveling salesman problems. We develop a mathematical model for the efficient and robust solving the traveling salesman problem (TSP). Based on the numerical study, the convergence properties of the model developed (i.e., the DNN-concept for solving TSP) is investigated. Ranges (or windows) of variation of the parameters of the developed mathematical model are determined (identified) to ensure (guarantee) the detection of the exact TSP solution/tour. In order to validate the mathematical model developed for solving TSP, a bifurcation analysis is carried out using the developed mathematical model. Various bifurcation diagrams are obtained numerically. The bifurcation diagrams obtained reveal the ranges of variation of some key parameters of the model developed to ensure (or guarantee) the convergence of the DNN-concept to the exact TSP-solution (i.e., global minimum). Concrete examples of graphs are considered and various numerical simulations are performed as proof of concept. Finally, a comparison of the results obtained with the results published in [17]-[18] lead to a very good agreement.