The optimization problem over a distributive lattice

In this paper we give a necessary and sufficient condition for existence of minimal solution(s) of the linear system A * X >= b where A, b are fixed matrices and X is an unknown matrix over a lattice. Next, an algorithm which finds these minimal solutions over a distributive lattice is given. Finally, we find an optimal solution for the optimization problem min {Z = C * X | A * X >= b} where C is the given matrix of coefficients of objective function Z.