Solving Ability of Hopfield Neural Network for QAP by Changing Chaotic Behavior of Switching Noise YoshifumiTada, Yoko Uwate and Yoshifumi Nishio Department of Electrical and Electronic Engineering, Tokushima University, Japan 2-1 Minami-Josanjima Tokushima 770-8506, Japan Email: {y-tada, uwate, nishio}@ee.tokushima-u.ac.jp Abstract—In our previous research, we confirmed that the chaotic switching noise generated by the cubic map gained a good performance for solving combinatorial opti- mization problems when the noise was injected to the Hop- field neural network. However, the reason of the good ef- fect of chaotic switching noise has not been clarified com- pletely. In this study, we investigate the solving ability of Hopfield neural network for QAP when the chaotic behav- ior of the switching noise is changed. 1. Introduction Combinatorial optimization problems can be solved with the Hopfield neural network (abbr. NN). If we choose con- nection weights between neurons appropriately according to given problems, we can obtain a good solution by the energy minimization principle. However, the solutions are often trapped into a local minimum and do not reach the global minimum. In order to avoid this critical problem, several people proposed the method adding some kinds of noise for solving traveling salesman problems (TSP) with the Hopfield NN [1]. Hayakawa and Sawada pointed out the chaos near the three-periodic window of the logistic map gains the best performance [2]. They concluded that the good result might be obtained by a property of the chaos noise; short time correlations of the time-sequence. Hasegawa et al. investigated solving abilities of the Hop- field NN with various surrogate noise, and they concluded that the effects of the chaotic sequence for solving opti- mization problems can be replaced by stochastic noise with similar autocorrelation [3]. We have also studied the reason of the good performance of the Hopfield NN with chaotic noise. We imitated the intermittency chaos by the burst noise generated by the Gilbert [4] model with 2 states; a laminar state and a burst state. We concluded that the irreg- ular switching of laminar part and burst part is one of the reasons of the good performance of the chaotic noise [5] [6]. Further, we have investigated a performance of chaotic switching noise generated by the cubic map when the noise is injected to the Hopfield NN for quadratic assignment problem (abbr. QAP). We have confirmed that the chaotic switching noise was effective for solving QAP similar to the intermittency chaos noise near the three-periodic win- dow [9]. However, the reason of the good effect of chaotic switching noise has not been clarified completely. In this study, we investigate solving ability of Hopfield NN for QAP when the chaotic behavior of the switching noise is changed. By computer simulation, we confirm that the network can find good solutions, even when the chaotic behavior is partly replaced by random time series. 2. Solving QAP with Hopfield NN Various methods are proposed for solving the QAP which is one of the NP-hard combinatorial optimization problems.The QAP is expressed as follow: given two ma- trices, distance matrix C and flow matrix D, and find the permutation P which corresponds to the minimum value of the objective function f (p) in Eq. (1). f (P )= N  i=1 N  j=1 C ij D p(i)p(j) , (1) where C ij and D ij are the (i, j )-th elements of C and D, respectively, p(i) is the i th element of vector P, and N is the size of the problem. There are many real applications which are formulated by Eq. (1). One example of QAP is find an arrangement of the factories to make a cost the min- imum. The cost is given by the distance between the facto- ries and flow of the products between the factories. Other examples are the placement of logical modules in a IC chip, the distribution of medical services in large hospital. Because the QAP is very difficult, it is almost impossi- ble to solve the optimum solutions in large problems. The largest problem which is solved by deterministic methods may be only 24 in recent study. Further, computation times is very long to obtain the exact optimum solution. There- fore, it is usual to develop heuristic methods which search near optimal solutions in reasonable time. For solving N -element QAP by Hopfield NN, N ×N neurons are required and the following energy function is defined to fire (i, j )-th neuron at the optimal position: E = N  i,m=1 N  j,n=1 w im;jn x im x jn + N  i,m=1 θ im x im . (2) NOLTA 2006 11-14 september Bologna - ITALY 107