Realization of Procedures Permuting Traveling Routes like the K-Opt by the Linked State Transition Neural Neworks for the Traveling Salesman Problems

Abstract

The Hopfield neural networks with continuous state transition have inefficient convergence in the neighborhoods of 0 or 1 as well as local convergence in their applications to the continuous versinos of 0-1 combinatorial optimization problems. Meanwhile, in the penalty approach to the constrained problems, it is feared that a lot of feasible combinatorial states satisfying the constraints corrupt into the local minima unconstrained problems. In order to settle these questions, “linked state transition neural networks with constraint satisfaction”, in which plural neurons transit simultaneously between discrete states so as to satisfy the constraints, are proposed from the standpoint of “discrete solutions to discrete problems by discrete neural networks”. In this paper, the linked state transitions in their applications to the traveling salesman problems realize automatical permuting procedures in a traveling route in the k-Opt method which is one of useful local search methods to combinatorial problems, and their numerical performance in experimental results are mentioned for relatively large-scale bench mark problems. These findings enhance practical values of the neural networks, and mediates between the fields of combinatorics and the neural networks closely.