I am trying to solve the below non-convex minimization problem
$$ \underset{X_1,x_2,X_3,\Lambda}{\mathrm{argmin}} \|A-X_1X_2\Lambda X_3 \|_F^2 $$
where $A \in \mathbb{R}^{n \times n}$, $X_1 \in \mathbb{R}^{n \times k}$, $X_2 \in \mathbb{R}^{k \times k_1}$, $X_3 \in \mathbb{R}^{k_1 \times n}$, and $\Lambda \in \mathbb{R}^{k_1 \times k_1}_{+}$ is a diagonal matrix. $A$ is a known matrix. I am trying to check if the above problem is strongly convex in each variable or not. I am not able to figure out how to check if the problem will be strongly convex in each variable. Is it possible to find constraints on the variables such that the problem becomes strongly convex in each variable?
Attempt1: I can rewrite the above problem as $$ \underset{X,Y,Z}{\mathrm{argmin}} \|A-XYZ \|_F^2 $$ and I want to check if the above function is convex in $Y$ or not. Let $G = \|A-XYZ \|_F^2$ and I vectorize $Y$, then the hessian of vec($Y$) can be written as $X^TX \otimes Z^TZ$. I don't how to find conditions on $X$ and $Z$ such that the hessian of $Y$ is positive definite.
The function cannot be strongly convex in each variable as the determinant of $X^TX \otimes Z^T Z$ is zero, the function is just convex.