Consider the following convex problem $$ \min_{\mathbf{x} \in C} f(\mathbf{x}) \ \text{subject to} \ G(\mathbf{x}) \in K. $$ We say strong duality holds if exists $\mathbf{x} \in C$ such that $0 \in \text{int}\{G(\mathbf{x})-K\}$, where $\text{int}$ denotes the interior of a set.
If $G(\mathbf{x})=\mathbf{a}^{\top}\mathbf{y}-b$ and $K=0$, where $\mathbf{a} \in \mathbb{R}^{n} \backslash 0,\mathbf{y} \in \mathbb{R}^{n}$. The set $C$ is defined by $C:=\{\mathbf{x}:\mathbf{l} \leq \mathbf{y} \leq \mathbf{u}\}$ is nonempty. Then if $0 \in \text{int}\{G(C )-K\}$ holds, thus strong duality holds. That is, $\mathbf{a}^{\top}\mathbf{y}-b=t$ has a solution for $\mathbf{y} \in C$, where $t$ sufficiently small. However, it does not always holds. Therefore, strong duality fails to holds. Does the conclusion right?