If I understand correctly, a given problem: \begin{equation} \min_{\mathbf{x}} \mathbf{c}^{\top}\mathbf{x}, \quad \mathbf{x} \in F \qquad \qquad (1) \end{equation}
is said to be a relaxation of $$ \min_{\mathbf{x}} \mathbf{c}^{\top}\mathbf{x}, \quad \mathbf{x} \in C \qquad \qquad (2) $$
if: $$ \inf{\{\mathbf{c}^{\top}\mathbf{x}: \mathbf{x} \in C\}} \geq \inf{\{\mathbf{c}^{\top}\mathbf{x}: \mathbf{x} \in F\}}. \qquad \qquad (3) $$
However, I would like to know what is the appropriate term for the opposite, i.e., having found a solution to equation (1) in the subset $\mathbf{x} \in F$, we need to reformulate the optimization such that $\mathbf{x} \in C$.
I am looking for the terminology here, which, for some reason, is not obvious in most literatures I have studied so far (which is admittedly not extensive, I am fairly new to this). Please help.
If $(1)$ is a relaxation of $(2)$, then $(2)$ is a restriction of $(1)$.
Also, you have the roles of $C$ and $F$ reversed in $(3)$.