A mathematical framework for solving dynamic optimization problems with adaptive networks

The paper develops a mathematical framework for solving dynamic optimization problems with adaptive networks (AN's) based on Hopfield networks. The dynamic optimization problem (DOP) includes a dynamic traveling salesman problem (TSP), in which the distance between any pair of cities in the conventional TSP is extended into a time variable. Compared to previous deterministic networks, such as the Hopfield network, the adaptive network has the most distinguished feature: it can change its states, continually reacting to inputs from the outside environment. From the scientific viewpoint, our framework demonstrates mathematically rigorously that the adaptive network produces as final states locally minimum solutions to the DOP. From the engineering viewpoint, it provides a mathematical basis for developing engineering devices, such as very large scale integration (VLSI), that can solve real world DOP's efficiently.