Subset of a bipartite graph with maximal number of minimal unmatched vertices
The problem is in $P$ by reducing the problem to submodular minimization. 

Let $v(A)$ be defined as the size of the maximum matching in $G_{|A}$. $v$ is submodular. One way to see this is convert the problem to a max flow problem, and use the fact that the set function $c(U)=$ max flow from $U$ to $t$ defined over $V\setminus \{t\}$ is submodular.

I will use $f(A)$ to mean the $f(G_{|A})$ in your notation.

$f(A) = |A| + |V| - 2v(A)$. $|V|$ is a constant, so we can maximize over $f(A)-|V| = |A| - 2v(A)$, which is the same as minimize $g(A) = 2v(A)-|A|$. $g$ is a submodular function(difference of submodular and modular function), and you can apply any polynomial time submodular function minimization algorithm.

This can be extended to weighted case if weighted analogue of $v(A)$ is submodular.