I am interested in solving the following optimization problem:
$$ \min_X \|XD-b\|_2^2 + \|X\|_* $$ Where $D \in\mathbb R^{n\times1}$, $ X \in\mathbb R^{n\times n}$ and $b \in\mathbb R^{n \times 1}$. That last term is the nuclear norm of $X$. I'm reading through An Accelerated Proximal Gradient Algorithm for Nuclear Norm Regularized Least Squares Problems. They developed a method that generalizes the problem above. Nonetheless, I could not understand some steps in their initial algorithm. I am going to explain their equations constructions in my words so you could capture any misunderstanding in my thoughts chain.
Assume that the first term in my equation is $f(X)$ and the last term is $P(X)$, they linearly approximate $f$ around $Y$ and then they added the term $\|X - Y\|_2^F$ as a Moreau-Yosida regularization (is it correct so far?) as follows:
Now, to get the optimum $X$, I would try to iterate and solve $\text{argmin}_{X} Q_\tau(X,Y^{t})$ and set it as $X^{t+1}$ and then assign $Y^{t+1} = X^{t}$ until it converges (i.e., no changes on $X^t$). However, their algorithm is constructed as follows:
My questions is why is in step 3 the $X_{k+1}$ constructed in this way (i.e., not $X_{k+1} = S_\tau(G)$)?