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.

  • $\begingroup$ That's not generally true. If A is positive semidefinite and $k$ is negative, you don't have equality. In addition to that, there is no solution to your problem as you have a strict inequality. $\endgroup$ – Johan Löfberg May 16 '18 at 10:46
  • $\begingroup$ Thank you Johan Löfberg, I edited my problem $\endgroup$ – user405592 May 17 '18 at 1:38

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