Finding probability from Dirichlet distributions under constraints

Question

Suppose that I have the following:

• Three Dirichlet distributions of dimension k: distr1, distr2, distr3
• A vector A of dimension 3 and a vector B of dimension k.

Does anyone know about an algorithm to find 3 probability vectors p1, p2 and p3 such that:

1. They satisfy the linear constraint A[0]*p1 + A[1]*p2 + A[2]*p3 = B
2. p1, p2 and p3 are in an area of high probability for distr1, distr2, distr3 respectively

(You can assume that the set of probability vectors p1, p2, p3 satisfying the linear constraint is not empty)

Any help or direction of thinking appreciated

thanks!