The 2-opt behavior of the Hopfield Network applied to the TSP

The Continuous Hopfield Network (CHN) became one of the major breakthroughs in the come back of Neural Networks in the mid 80s, as it could be used to solve combinatorial optimization problems such as the Traveling Salesman Problem. Once researchers provided a mechanism, not based in trial-and-error, to guarantee the feasibility of the CHN, the quality of the solution was inferior to the ones provided by other heuristics. The next natural step is to study the behavior of the CHN as an optimizer, in order to improve its performance. With this regard, this paper analyzes the attractor basins of the CHN and establishes the mathematical foundations that guarantee the behavior of the network as a2-opt; with the aim to open a new research line in which the CHN may be used, given the appropriate parameter setting, to solve ak-opt, which would make the network highly competitive. The analysis of the attraction basins of the CHN and its interpretation as a2-optis the subject of this article.