I want to know if there is a closed form expression for the optimal objective function? How can I get it, if it does exist?
Condition: $h,f\in \mathbb{C}^{N\times1}, \epsilon > 0 $.
$\max \ \ e^Hhh^He + (f^Hhh^H-h^H)e + e^H(hh^Hf-h) \\ s.t.\ \ e^He \leq \epsilon^2 $
This is not an answer. I was running out of space in comments. It is not difficult to recast your problem as \begin{align} \max_{\mathbf{x}}&~\mathbf{x}^T\mathbf{h}\mathbf{h}^T\mathbf{x}-\alpha \mathbf{h}^T\mathbf{x} \\\ &\mathbf{x}^T\mathbf{x}\leq 1 \end{align} where $\alpha$ is some suitable real constant. This is clearly a convex problem. The $\mathbf{h}$ in this is not same as the $\mathbf{h}$ given in your problem. $\mathbf{h}=[\mathbf{h}_R^T,\mathbf{h}_C^T]^T$ where $\mathbf{h}_R$ and $\mathbf{h}_C$ are the real and complex part of your original $\mathbf{h}$. Now applying KKT is bit easier. I have posted it as a question here. Find the real vector $x$ which satisfies all this? . Any solution to that question should address your problem.