In this paper, convergence theorems of discrete Hopfield-type neural networks with delay(DNND) are obtained.It is shown that networks will reach a stable state at most 2n+1 steps after energy function associated with the network has converged, whereas discrete Hopfield-type neural networks reach a stable state at most n+2 steps. We present delay-perturbation algorithm to solve combinational optimization problems by DNND and clarify nature relations between DNND and DNN networks, thus we construc...