MINIMUM-SEEKING PROPERTIES OF ANALOG NEURAL NETWORKS WITH MULTILINEAROBJECTIVE FUNCTIONS

In this paper, we study the problem of minimzing a multilinear objecti ve function over the discrete set {0,1}(n). This is an extension of an earlier work addressed to the problem of minimizing a quadratic funct ion over {0,1}(n). A gradient-type neural network is proposed to perfo rm the optimization, A novel feature of the network is the introductio n of a so-called bias vector, The network is operated in the high-gain region of the sigmoidal nonlinearities, The following comprehensive t heorem is proved: For all sufficiently small bias vectors except those belonging to a set of measure zero, for all sufficiently large sigmoi dal gains, for all initial conditions except those belonging to a nowh ere dense set, the state of the network converges to a local minimum o f the objective function, This is a considerable generalization of ear lier results for quadratic objective functions, Moreover, the proofs h ere are completely rigorous, The neural network-based approach to opti mization is briefly compared to the so-called interior-point methods o f nonlinear programming, as exemplified by Karmarkar's algorithm, Some problems for future research are suggested.