Be a discrete nonlinear convex optimization problem $P$
\begin{align} \underset{x\in \mathrm{C}^n}{\mathrm{min}} \ \ \ f(x) \\ Ax=b \\ c \leq x \leq d \end{align}
$C$ is a dense in $F$.
Is solving $P$ in $C$ equivalent to solving $P$ in $F$, since for any optimal solution $x^F$ found in $F$, we can find a solution $x_m^C$ arbitrarily close to $x^F$ as $m\rightarrow +\infty$.
I understand that the equality constraint are not necessary satisfied with $x_m^C$. But $Ax$ can be made arbitraly close to $b$ by taking $m$ large enough.
What books/articles do you recommend that deals with this kinf of relaxation?
Thanks in advance