Finding a coefficient matrix to maximize a given variable under some constraints

Question

Considering a linear system of the form:

Is there a method to find the coefficient matrix such that $\mu_F$ is maximal/minimal under some constraints (e.g., $0<c_j^i<1$ and $\sum_{i}c_j^i=1$ $\forall$ $i,j$)?

Note that we can assume the matrix is non-singular.