I am interested in a sequence of linear programs of the following type, over $x\in\mathbb{R}^k$: $$ \min c_n^T x\\ x\geq 0\\ Ax = b $$ for $x\in\mathbb{R}^k$, $n\geq 1$ and some choice of $A, b$ that remain fixed. The vector $c_n$ is given by $$ c_n = c_1\circ c_1\circ\cdots\circ c_1 \hspace{1cm}(n\text{ times})$$ where $\circ$ is entrywise multiplication. That is, $c_n = (c_1)^{n}$ is the $n$-th pointwise power of some vector $c_1$. The vector $c_1$ is nonnegative off the bat.
My question is: Under what conditions on $c_1$ (or $A$, $b$) does it happen that a solution $x_n$ to the $n$-th such problem is also a solution to the $(n+1)$-st such problem? It would also be helpful if someone can point me to the proper terminology that might help me identify this in the literature on linear programs, so I can read more about it there. I'm not sure if this is something that has been studied extensively or not.