I am studying convex optimization from Boyd. In the problem 5.22 I need to find the dual function of the optimization problem
\begin{equation*} \begin{aligned} & \underset{x}{\text{minimize}} & & e^{-x} \\ & \text{subject to} & & \frac{x^2}{y} \leq 0 \end{aligned} \end{equation*}
I found that the dual function is $$g(x,y,\lambda) = \inf_{x,y >0} (e^{-x} -\lambda\frac{x^2}{y}).$$ Now, I am trying to see what is the infimum and when it is achieved. I was wondering if any could help me computing this or give some general guidelines how to go about this, I always get stuck in this part of problems like this. Thanks!
You are minimizing $e^{-x}$ which does not depend on $y$, so your constraint does not impose any restrictions, just pick any $y \le 0$, and minimum $e^{-x}$, which has a minimum of $0$ as $x \to \infty$...