NEUROCOMPUTINC Neurocomputing 14 (1997) 139-156 Associating arbitrary-order energy functions to an artificial neural network Implications concerning the resolution of optimization problems G. Joya *, M.A. Atencia, F. Sandoval Depurtemenfo Tecnologia Electrcinica. Campus Teatinos, Uniuersidad de Mklaga, 29071 Mu’laga, Spain Received 6 February 1995; accepted 3 1 January 1996 zyxwvutsrqponmlkjihgfedcbaZYX Abstract We have studied the restrictions that a first order asynchronous feedback neural network must fulfill to be associated to an arbitrary order energy function of the kind described by Kobuchi [6], i.e., the network evolution is related to the descent to a minimum of such a function. These restrictions do not avoid the association of the even order energy functions to a first order network. However, for the odd order energy functions, most of the weights of each neuron must be zero. This result discards using first order neural networks for the solution of optimization problems associated with an odd order function, justifying in this way the use of high order neural networks. For these ones, we have obtained a general expression of their possible energy functions, which includes, as a special case, the high order generalization of Hopfield’s energy functions until now used, for example, in [5], [8]. Keywords: Asynchronous feedback neural network; Optimization problem; Energy function; High order neur,xl network; Multinomial function order 1. Introduction An artificial neural network designed to work as an associative memory or to solve an optimization problem is a dynamic system, whose convergence and stability must be * Corresponding author. Email: joya@tecmal .ctima.uma.es. 0925-2312/97/$17.00 Copyright 0 1997 Elsevier Science B.V. All rights reserved PII SO925-2312(96)00033-l