I want to solve the following approximation error problem $$\min_{x\in S}\|Ax-b\|_p$$. Here $A$ is in $\mathbb{R}^{n\times d}$, $b$ is in $\mathbb{R}^n$ and $x$ is in $\mathbb{R}^d$. $n \gt d$ and $p$ is a norm between $1 \leq p \leq \infty$
$S$ is a compact set in $\mathbb{R}^d$. This would ensure that the minimization always exists.
Suppose I know $b$ is bounded in the $p$ norm.
Can I say that the value of $$\min_{x\in S}\|Ax-b\|_p$$ would also be finite?