Otherwise, it represents an invalid path. In the transposition matrix, only one element in each row and column can be 1, and the rest are 0. Where Represents the element in row X and column i of the transposition matrix, and when its value is 1, it means that step i accesses city x, and when its value is 0, it means that step i does not access City X.
If the subscripts x and y represent the city and i represent the ith visit, the path length can be expressed in the following general form: In TSP problem, the energy function often reflects the length of the path. Where Is a penalty function for violating constraints, Is the objective function of optimization And For balance and The proportional constant of the action in the total energy function, and if minimized, then ； If maximization, then ；.
Lagrange function and sub multiplication or penalty function method in optimization theory are usually used to construct energy function (formula (1)). Another key problem is to construct the energy function so that its minimum value corresponds to the optimal solution of the problem, which determines whether a specific problem can be solved by neural network. Generally, each feasible solution of the problem can be expressed by transposition matrix.
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The key of using neural network to solve the optimization problem is how to map the optimization problem to be solved into a neural network. Therefore, after comprehensive judgment, continuous Hopfield neural network is selected as the algorithm model of this problem. TSP problem is a typical combinatorial optimization problem. According to its characteristics, discrete Hopfield neural network is usually used to deal with associative memory problems, and continuous Hopfield neural network is usually used to deal with combinatorial optimization problems. Continuous Hopfield neural network not only simplifies the properties of biological neurons, but also accurately retains the dynamic and nonlinear characteristics of biological neural network. Changing the state of neurons according to the dynamic rules will eventually make the neural network in a stable state.ĭiscrete Hopfield neural network is a two-state nonlinear dynamic system with multiple input and multiple output generation thresholds. Because the network associative memory process of Hopfield neural network is the process that the nonlinear dynamic system runs towards a stable state. Continuous Hopfield neural network is a continuous time neural network model, In this network, the state of neurons can take any real value between 0 and 1. Neurons in discrete Hopfield neural network are quite different from biological neurons because the input and output of biological neurons are continuous and have time delay. Hopfield neural network is mainly divided into discrete Hopfield neural network (DHNN) and continuous Hopfield neural network (CHNN).
Hopfield neural network (HNN) is a fully interconnected feedback neural network, in which each neuron is connected with other neurons.
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