Let $A$ be a $n\times n$ hermitian matrix. My goal is to minimize the following hermitian form with an additionnal "real constraint":
$$\min_f f^\ast A f$$ $$\text{subject to }f \in \mathbb{R}^n$$
This optimization problem can be solved easily without the constraint $f\in \mathbb{R}^n$ (singular value decomposition).
Is there a general way to handle this problem ? I think that the best solution could be to use a gradient-descent scheme but I do not see any reason for $f\in \mathbb{R}^n \to f^T A f \in \mathbb{R}$ to be convex.