Solving 50-City TSP Problem Using Hopfield Neural Network

Solving 50-City TSP Problem Using Hopfield Neural Network

Hopfield Neural Network is a type of recurrent neural network particularly suitable for solving combinatorial optimization problems like the famous Traveling Salesman Problem (TSP). The TSP requires finding the shortest possible route that visits each city exactly once and returns to the origin city. For a 50-city TSP problem, traditional methods involve enormous computational complexity, while Hopfield Neural Network provides an efficient heuristic solution approach.

### Fundamental Principles Neuron Representation: Each neuron corresponds to a city's position in the route. For example, neuron V_xi represents whether city x is visited at the i-th position in the tour. In code implementation, this is typically represented as a 50×50 binary matrix where rows represent cities and columns represent positions. Energy Function: The Hopfield network converges to a stable state by minimizing an energy function. For TSP, the energy function must satisfy three constraints: - Each city must be visited exactly once - Each position in the route must contain exactly one city - The total route length should be minimized Weight Design: Connection weights between neurons encode these constraints. For instance, inhibitory weights prevent the same city from appearing in different positions or multiple cities occupying the same position. The weight matrix W_xi,yj typically contains terms for distance penalties and constraint violations.

### Implementation Steps Initialization: Randomly initialize neuron states representing an initial route. In code, this involves creating a random permutation matrix or using small random values around 0.5. Iterative Update: Adjust neuron activation values based on the energy function's gradient using the update rule: V_xi(t+1) = f(Σ W_xi,yj × V_yj(t) + bias), where f is typically a sigmoid or step function. This gradually optimizes the route through multiple iterations. Convergence Check: Stop the algorithm when the energy function shows no significant decrease or when maximum iterations are reached, then output the final route. Convergence is typically monitored by checking energy change below a threshold (e.g., 1e-6) over consecutive iterations.

### Advantages and Limitations Advantages: Well-suited for small to medium-scale TSP problems; exhibits significant parallel computation potential through matrix operations. Limitations: May converge to local optima; requires careful manual tuning of hyperparameters like weight coefficients and learning rates. The performance heavily depends on proper parameter selection for constraint terms and distance terms in the energy function.

For beginners, understanding the energy function design and physical meaning of neurons is crucial. Subsequent improvements can incorporate strategies like simulated annealing to enhance solution quality by helping escape local minima through temperature-controlled randomization.