Consider the objective function
$$\min_{X}\|X_{(1)}\|_{*}+\|X_{(2)}\|_{*}+\|X_{(3)}\|_{*}+\lambda\|Ax-b\|_2^2$$
where $X$ is a three order tensor, $X_{(i)}$ is a matrix whose columns are the mode-$i$ fibers of $X$(i=1,2,3), $x$ is vec($X$), $\lambda$ is constant, and $\|\cdot\|_*$ is the nuclear norm.
How to introduce auxiliary variables to make the objective function separable?
\begin{array}{ll} \text{minimize} & \|Z_1\|_* + \|Z_2\|_* + \|Z_3\|_* + \lambda\|A\mathop{\textrm{vec}}(X) - b \|_2^2 \\ \text{subject to} & Z_1 = X_{(1)} \\ & Z_2 = X_{(2)} \\ & Z_3 = X_{(3)} \end{array}