We are looking for any results on the effective computability of the optimization algorithms.
In particular, consider probability mass functions on a finite set $X=\{x_1, ... x_n\}$.
We are looking for a probability mass function that satisfies a finite set of linear inequalities of the form $\sum_{j=1}^{k_i} P(x_{i_j}) \leq c_i$ and minimizes the negative Shannon entropy $H(P)= \sum_{i=1}^{n} P(x_i) \log(P(x_i))$. We know that this has a unique solution.
We want to compute this unique solution or approximate it with any given arbitrary precision.
I was wondering if there is an effectively computable algorithm for solving this.
Thanks a lot in advance!