This is related to a previous post, see here if needed.
I have $x \in \mathbb{R}^{d\times 1}$ and $y\in \mathbb{R}^{p\times d}$ and I want to minimize the function $f(x,y) = g(x) + x^T y^T y x$ such that $y^T y$ is a positive definite matrix and $g$ is a strongly convex function and has $L$-Liptschitz continuous gradient.
- Is there a guarantee that the following function admits a minimum? If not, what other assumptions/conditions are needed to ensure the function has a minimum?
I think since $\nabla^2_{xx} f(x,y)$ is positive definite, then the function should have a minimum in the $x$-direction but not sure if this guarantees that an optimal point $(x,y)$ exists.
- Additionally, do I need to impose $a \leq \|y\| \leq b$ where $a>0$, to ensure the minimization problem doesn't converge to a trivial solution $y=0$?
I think this is not strictly necessary for ensuring the existence of a minimum but rather to avoid trivial solutions.