I am solving problem 5.22 b) from Boyd and Vanderberghe "Convex Optimization". The problem is that given the optimization problem $$\begin{equation*} \begin{aligned} & \underset{x}{\text{minimize}} & & x \\ & \text{subject to} & & x^2 \leq 0\\ \end{aligned} \end{equation*}$$
I need to find the dual and the primal and dual optimal points. I found that the dual problem is
\begin{equation*} \begin{aligned} & \underset{x}{\text{maximize}} & & -\frac{1}{4\lambda} \\ & \text{subject to} & & \lambda>0 \end{aligned} \end{equation*}
And from this, I concluded hat the optimal point is $p^* = 0$ for $x^* = 0$. However, my conclusion for the dual was that we could not attain the dual optimal point. Thus, there is strong duality HOWEVER, the solutions say that the dual optimal is $d^* = 0$ but it is not achieved and so there is strong duality. I don't understand why the answer gives you an optimal dual $d^*$ and then says you can't achieve it. If you can't achieve it, then why to give it? Also, if we can't achieve it, how do we have strong duality? Maybe I did something wrong when computing the dual problem? Thanks for your help!