Theoretical Exploration on Local Stability of Simultaneous Recurrent Neural Network Dynamics for Static Combinatorial Optimization

Abstract— This paper presents a theoretical local stability analysis of the Simultaneous Recurrent Neural network (SRN) as a nonlinear dynamic system operating in relaxation mode for static combinatorial optimization. Specifically, stability of hypercube corners of the SRN dynamics, which are equilibrium points for high-gain node dynamics and useful entities to represent solutions of combinatorial optimization problems, is examined. A theorem that exposes existence of network weights to establish hypercube corners as stable equilibrium points is stated. Theoretical insight obtained in this study combined with extensive simulation based empirical evidence reported in the literature to date suggests that the SRN dynamics operating in relaxation mode appears to possess desirable stability characteristics as a recurrent neural network for addressing combinatorial optimization problems.