The joint optmization problem is
$\text{minimize}_{\mathbf{X}, t} \quad \text{trace}(\mathbf{X}) \\\text{subject to} \quad \text{trace}(\mathbf{A}\mathbf{X})\geq t \\ \quad\quad\quad\quad\quad X \succeq 0, t \geq k.$
where $\mathbf{A}$ is positive semidefinite matrix and $k \geq0$.
Intuitively, I know that objective function minimized when t=k(minimum of t) because feasible set of $\mathbf{X}$ is expanded. But I can not know specific reason/proof. please explain specific reason.