There are different ways to optimize portfolios:
$$ \max R^Tw\tag{1}$$
or
$$ \min w^T \Sigma w\tag{2}$$
and finally using a risk tolerance $\lambda$:
$$ \min{(w^T\Sigma w-\lambda R^T w)}\tag{3}$$
Suppose we have the constraint $\sum w_i = 1$, $w_i\ge 0$ for all the optimization problems.
Additionally, we can define further constraints for problems $(1)$ and $(2)$:
For $(1)$: $w^T\Sigma w\le \sigma$, i.e. the risk should not exceed a certain level $\sigma$.
The same is possible for $(2)$ with return, adding the constraint: $R^T w\ge r$, for a minimal target return $r$.
My question is, in the optimization problem $(3)$, does it make sense to add a constraint like $w^T \Sigma w \le \sigma$ or $R^Tw \ge r$?
Am I right to say that adding such a constraint we would discard the solution (a efficient frontier portfolio) which does not satisfy this constraint?