Abstract Matroids of rank n are studied in which each element has a real-valued cost and one of d > 1 colors. The problem of finding a minimum cost base in the matroid subject to linear inequality constraints on colors is explored. The color constraints are shown to form a strict hierarchy based on increasingly stronger notions of convexity. The concept of a lattice of color vectors and associated minimum cost bases is introduced. The relationship of the cost of a base to those of its neighbors in the lattice is examined. It is shown that the solution to the constrained problem must occur at constraint boundaries, allowing earlier algorithms for a simpler version of the problem to be extended. Finally, it is shown that a given set of constraints can be located within the hierarchy in polynomial time.