Here is question and solution,but i didn't understand the solution,can anyone help me to understand it?
min $-3x_1^2+x_2^2+2x_3^2$
$s.t. x_1^2+x_2^2+x_3^2=1$
Does the strong duality hold?
Solution:
Let A=\begin{bmatrix} -3 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \end{bmatrix}and $X=[x_1,x_2,x_3]^T$,so now rewrite as
min $X^TAX$,so its lagragian function is $X^TAX+\mu(X^TX-1)$,so
dual problem is
max $-\mu$
$s.t. A+\mu I \ge 0$
$ A+\mu I \ge 0$,so $\mu \ge -\lambda\ min(A)=3$,that is $d^*=-\mu=-3$
$P^*=\lambda\ min(A)=-3$,so $d^*=P^*$,it means strong duality holds
My question is
$1.$ Where is the $d^*$ and $P^*$,because i don't see them in the solution formula and question,i know they are about the Slater's condition,but i don't know what is their meaning in this solution,why when $d^*=P^*$ means strong duality holds?
$2.$ About this formula,$ A+\mu I \ge 0$,so $\mu \ge -\lambda\ min(A)=3$,where is $\lambda\ min(A)$ from? it just appeared suddenly