Minimum-seeking properties of analog neural networks with multilinear objective functions

In this paper, we study the problem of minimizing a multilinear objective function over the discrete set {0, 1}ⁿ. This is an extension of an earlier work addressed to the problem of minimizing a quadratic function over {0, 1}ⁿ. A gradient-type neural network is proposed to perform the optimization. A novel feature of the network is the introduction of a so-called bias vector. The network is operated in the high-gain region of the sigmoidal nonlinearities. The following comprehensive theorem is proved: For all sufficiently small bias vectors except those belonging to a set of measure zero, for all sufficiently large sigmoidal gains, for all initial conditions except those belonging to a nowhere dense set, the state of the network converges to a local minimum of the objective function. This is a considerable generalization of earlier results for quadratic objective functions. Moreover, the proofs here are completely rigorous. The neural network-based approach to optimization is briefly compared to the so-called interior-point methods of nonlinear programming, as exemplified by Karmarkar´s algorithm. Some problems for future research are suggested