  Given the single-letter capacity formula and the converse proof of a channel
without constraints, we provide a simple approach to extend the results for the
same channel but with constraints. The resulting capacity formula is the
minimum of a Lagrange dual function. It gives an unified formula in the sense
that it works regardless whether the problem is convex. If the problem is
non-convex, we show that the capacity can be larger than the formula obtained
by the naive approach of imposing constraints on the maximization in the
capacity formula of the case without the constraints.
  The extension on the converse proof is simply by adding a term involving the
Lagrange multiplier and the constraints. The rest of the proof does not need to
be changed. We name the proof method the Lagrangian Converse Proof. In
contrast, traditional approaches need to construct a better input distribution
for convex problems or need to introduce a time sharing variable for non-convex
problems. We illustrate the Lagrangian Converse Proof for three channels, the
classic discrete time memoryless channel, the channel with non-causal
channel-state information at the transmitter, the channel with limited
channel-state feedback. The extension to the rate distortion theory is also
provided.
