This paper studies the global minimum conditions of the outer supervised feedforward neural networks （FNN） in the batch style and sequence style trainings.For the sequence style training

it is showed that there always exist N local minimum points in the error surface.For the batch style training

it is showed the sufficient and necessary condition that FNNs can obtain global minimum solutions with null cost is that the range space R（Y） of the outer supervised signal matrix Y must be included in the range space R（X） of the hidden output matrix X

i.e.

R（Y）R（X） .In addition

it is also showed that the sufficient condition that the FNN can obtain global minimum solutions with null cost is M≥N .It should be specificially stressed that under the conditions of R（Y）R（X） and M≥N

there will exist not any local minimum points in the error surface.Further

it follows that if C≤M<N it is possible for the FNNs to obtain global minimum solutions with null cost

if M<C≤M it is impossible for the FNNs to obtain global minimum solutions with null cost.