OPTIMIZATION BY USING A MODIFIED HOPFIELD NETWORK

The paper presents a novel method for solving optimization problems by using a modified Hopfield network. In a conventional Hopfield network weight values of the net are calculated before simulation and are constant. Our method is a novel method, because the values of input signals or weights are modified during simulation. The method makes use of the Hopfield net with continuous function of neurons according to Eq. (2). The model Hopfield net in electronic components is shown in Figure 1. An energy function of the neural net is described by Eqo (3). Comparing Eq. (3) and Eq. (4), which is a general form of an optimization problem cost function, we get weight and external input signal values. The Hopfield net is implemented in software in this work. A simulating program makes use of Eq. (11) to calculate the input of each neuron in the net. During the simulation the input signals are modified in accordance with Eq. (12). The duration of input signals modifying is defined by a random value nnar. Finally, a number of iterations to achieve a stable state of the net are done. A number of trials are performed for each optimization problem and the best results are chosen. Figure 2 shows the algorithm of the method. In this work, simulations were done for six examples of the travelling salesman problem. A cost function of the travelling salesman problem is described by Eq. (17). This function consists of four components: the total length of the salesman's tour, two terms, which ensure that the salesman's tour is valid and the term, which forces neurons to have the output signal equal zero or one. The method with input signal values modifying during simulation gives better results than the conventional one. Results are performed in Table 1 and 2. For ten-city problems the modified Hopfield net finds the optimal solution with 100% success rate. Some conclusions coming from using this method are presented.