Consider the following function: $$ F(X_1,X_2,X_3,X_4) = \|X_1 X_3^{\top} - Y_1\|_F^2 + \|X_2 X_3^{\top} - Y_2\|_F^2 + \|X_2 X_4^{\top} - Y_3\|_F^2 + \lambda \sum_{i=1}^4 \|X_i\|_F^2, $$ where $X_1 \in \mathbb{R}^{n_1 \times r}$, $X_2 \in \mathbb{R}^{n_2 \times r}$, $X_3 \in \mathbb{R}^{n_3 \times r}$ , and $X_4 \in \mathbb{R}^{n_4 \times r}$. The dimensions $n_1, n_2, n_3, n_4$ are different.
I know that when considering each of the matrices separately (fixing all others), $F$ is strongly convex in each of these block coordinates. But I am not sure whether it is convex (at all) in all of its coordinates simultaneously.
Edit: I forgot to mention that $\|\cdot\|_F$ is the Frobenius norm and $Y_1,Y_2$, and $Y_3$ are given matrices of appropriate dimension.