There is a pressing need to verify robustness of classification deep neural networks (CDNNs) as they are embedded in many safety-critical applications. Existing robustness verification approaches rely on computing the over-approximation of the output set, and can hardly scale up to practical CDNNs, as the result of error accumulation accompanied with approximation. In this paper, we develop a novel method for robustness verification of CDNNs with sigmoid activation functions. It converts the robustness verification problem into an equivalent problem of inspecting the most suspected point in the input region which constitutes a nonlinear optimization problem. To make it amenable, by relaxing the nonlinear constraints into the linear inclusions, it is further refined as a linear programming problem. We conduct comparison experiments on a few CDNNs trained for classifying images in some state-of-the-art benchmarks, showing our advantages of precision and scalability that enable effective verification of practical CDNNs.