Definition: Given a matrix $X\in\mathbb{R}^{m\times n}$ and a positive integer $r<\min\{m,n\}$, the truncated nuclear norm $\|X\|_{r,*}$ is defined as the sum of $\min\{m,n\}-r$ minimum singular values, that is, $\|X\|_{r,*}=\sum_{i=r+1}^{\min\{m,n\}}\sigma_{i}(X)$ where $\sigma_{i}(X)$ is the $i$th singular value of $X$. (Hu et al., 2013)
Let $X$ and $Z$ be $m\times n$ matrices. I am trying to solve the following optimization problem:
\begin{equation} \hat{X}=\operatorname{arg}\min_{X}~~\alpha\|X\|_{r,*}+\frac{1}{2}\beta\|X-Z\|_{F}^{2} \end{equation} where $\|\cdot\|_{F}$ denotes the Frobenius norm.
I know this problem is nonconvex, but is it possible to get the global optimal solution? If yes, how? Thanks in advance for your help in any way.