Excuse me if my question is stupid. I'm seeking the references on the dependence of the (linear) optimization problem on (linear) constraints. Namely, consdier the following optimization problem:

$$P(\alpha)~~:=~~\sup_{p\in\mathcal A(\alpha)}~c^T p,$$

with

$$\mathcal A(\alpha)~~:=~~\big\{p\in\mathbb R^n:~ Ap~=~\alpha\big\},$$

where $\alpha\in\mathbb R^m$, $c\in\mathbb R^n$ and $A\in\mathbb R^{m\times n}$ are given. I'm looking for the references on the map $\alpha\longrightarrow P(\alpha)$. Any suggestions and comments are highly appreciated. Thanks a lot!

PS: Thanks for the reply. Indeed, here we suppose of course that $m>n$ and actually $p_i\ge 0$ and $\sum_{i=1}^np_i=1$. So if the set $\mathcal{A}(\alpha)\neq \emptyset$, then the maximizer always exists.