Orthogonal projections and the assignment problem

The neural network approach to optimization problems, such as the assignment problem (AP) and the travelling salesman problem (TSP), has introduced new representations. The paper presents the theoretical explanation of the feasible space, and the simplification of the dynamics of the neural approach to the AP brings together several results and observations in the literature on this subject. We have also applied these observations to the TSP in which we projected the activation vector onto a feasible space after each dynamical update. This technique significantly improves the performance of the Hopfield approach, but the results are not very good in comparison to traditional heuristics such as the 2-opt. The most far reaching result of this paper is the possibility of designing networks in the frequency domain by using the appropriate Fourier decomposition. Our approach to the AP and TSP is just one instance of such a methodology.