Quadratic programming: Approximate Solution

Question

Let $A$ be a $p \times p$, positive definite and symmetric matrix and $t \in \mathbb{R}^p$, such that $t_i>0$ for at least one $i \in \{1,\dots,p\}$. Let $x^*$ be the unique minimizer of $$min \{x'Ax: x \geq t\}. $$ Let $t_n$ be a sequence in $\mathbb{R}^p$ such that $n t_n \to c \in \mathbb{R}^p$ as $n \to \infty$. Let $x_n^*$ be the unique minimizer of $$min \{x'Ax: x \geq t+t_n\}. $$ In general, $x_n^* \neq x^* + t_n$, but is it true that $n(x_n^* - x^*) \to c$ as $n \to \infty$? If so, how can I show that?