Suppose you have an equation of the form $Ax=f(x)$, where $A$ is a $n \times n$ matrix, $x$ is a vector of length $n$ and $f(\cdot)$ is some function. Is there a name for this sort of problem?

If $f$ is assumed to be small in some sense, a common choice is semilinear equation. 


and to be sure: the notion of "nonlinear eigenvalue problem" is usually used for equations of the form $A(u)=\lambda u$, where $A$ is a nonlinear operator (or a nonlinear mapping from $R^n$ to $R^n$, if you wish). 


Do you have any examples in mind? As it stands it is essentially a system of $n$ simultaneous equations in $n$ variables. That term would (as commented above) usually suggest a system $$g_i(x)=0 \text{ for } 1 \le i \le n.$$ But that is easily obtained by setting $A$ to be the zero matrix. If that is too degenerate then we can use your formulation with $A$ your favorite matrix and $f(x)$ returning the vector whose $i$th component is $g_i(x)+(Ax)_i.$ So some restriction is needed to get an interesting question. 


This does not answer the question as stated, but your specific question (assuming the inverse is coordinatewise) can be stated as $\min_i \min x_i,$ subject to the system of constraints $x_i <A_i, x> = 1.$ If you remove one level of $\min,$ you are trying to minimize the function (wlog) $x_1$ over the set defined by the constraints, which reduces to a lagrange multiplier problem: $1 = \lambda_1 \langle A_1, x\rangle + \sum_{j=2}^n \lambda_j A_{j1} x_j.$ and for $k=2, \dotsc, n$ $0 = \lambda_1 x_1 A_{1k} + \lambda_k \langle A_k, x\rangle + \sum_{j=2, j \neq k}^n A_{jk} x_1.$ Unless there is some trick to reduce this system (quite possible, I don't really want to think too hard about it), this is a general system of quadratic equations, which is ETR hard (that is, about as hard as any computational problem), so in that sense, @Quiochu's comment is very a propos. 

