Suppose we have an optimization problem of the form

$\max_{A,p}f(A)^T p$

s.t.

$Ap=c$

$\sum_{i=1}^n p_i = 1$

Where $A \in \mathbb{R}^{n\times n}$, $p \in \mathbb{R}^n$, and $f(A) : \mathbb{R}^{n\times n} \mapsto \mathbb{R}^n$ and is linear. As stated, this appears to be a Quadratically Constrained Quadratic Program.

But suppose it is known that the space of full rank A contains an optimal solution. If we were to optimize over that restricted space, only a single $p$ satisfies the constraint for each $A$. So in that case it seems unnecessary to optimize over both $A$ and $p$ when a feasible assignment of $A$ determins a single feasible assignment to $p$.

What I'm wondering is whether there is anything to be gained by reformulating the problem to eliminate $p$, something like

$\max_{A \in \{\text{full} \; \text{rank} \;nxn \; \text{matrices}\}} f(A)A^{-1}c$

s.t.

$\sum_{i=1}^n (A^{-1}c)_i = 1$

What problem class does the latter formulation belong to, and does the transformation admit a more efficient solution? I'm new to optimization, so please excuse my ignorance on the subject.