Some results on the Sign recurrent neural network for unconstrained minimization

The sign dynamical system for unconstrained minimization of a continuously differentiable function f is examined in this paper. This dynamical system has a discontinuous right hand side and it is interpreted here as neural network. Asymptotic convergence is proven (by using Filippov's approach) finite-time convergence of its solutions is established and an improved upper bound for convergence time is given. A first contribution of this paper is a detailed calculation of Filippov set-valued map for the sign dynamical system, in the general case, i.e. without any restrictive assumptions on the function f to be minimized. Convergence of its solutions to stationary points of f follows by using standard results, i.e. a generalized version of LaSalle's invariance principle. Next, in order to prove finite-time convergence of solutions, the applicability of standard results is extended so that they can be applied to the sign dynamical system. Finally, while establishing finite-time convergence, a novel proving procedure is introduced which (i) allows for milder assumptions to be made on the function f, and (ii) results in an improved upper bound for the convergence time. Numerical experiments confirm both the effectiveness and finite-time convergence of the sign neural network.