  Given an original discrete source X with the distribution p_X that is
corrupted by noise to produce the noisy data Y with the given joint
distribution p(X, Y). A quantizer/classifier Q : Y -> Z is then used to
classify/quantize the data Y to the discrete partitioned output Z with
probability distribution p_Z. Next, Z is transmitted over a deterministic
channel with a given channel matrix A that produces the final discrete output
T. One wants to design the optimal quantizer/classifier Q^* such that the cost
function F(X; T) between the input X and the final output T is minimized while
the probability of the partitioned output Z satisfies a concave constraint
G(p_Z) < C. Our results generalized some famous previous results. First, an
iteration linear time complexity algorithm is proposed to find the local
optimal quantizer. Second, we show that the optimal partition should produce a
hard partition that is equivalent to the cuts by hyper-planes in the
probability space of the posterior probability p(X|Y). This result finally
provides a polynomial-time algorithm to find the globally optimal quantizer.
