Made a crucial mistake in the problem formulation; please delete.
Are these really all the constraints? Let $u$ be the vector of all $1$'s. Consider solutions of the following form: $p$ is an arbitrary vector with $u^T p = 1$,
$A = B + c u^T$ where $B p = 0$. If your problem has an optimal solution, $f(B)^T p$ must be $0$ for all $B$ such that $B p = 0$.