Given a matrix $A$, each element $A_{i,j} \geq 0$, find the vector $\vec x$ that maximizes the minimum element in $\vec b$ ($\vec b = A \vec x$). Note that this is not a linear equation system as I don't know $\vec b$.

Extra contraints on the solution are $x_i \geq 0$, and $\sum x_i = 1$.

Is this possible to solve, and if so, how? Can it have 0 or more than one solution?