I am going over this paper on transaction costs optimisation for online portfolio selection and am having trouble with the proof of Proposition 4.2. Specifically, they write \begin{align*} \tilde{\mathbf{b}}_{t+1}={\arg\min}_{\tilde{\mathbf{b}}}f(\tilde{\mathbf{b}})+\lambda\Omega(\tilde{\mathbf{b}}) \end{align*} where $f$ is a differentiable convex function and $\Omega$ is the L1 norm. They then claim that a closed form can be found by solving the following \begin{align*} \begin{cases} \tilde{\mathbf{b}}_{t+\frac{1}{2}}=\arg\min_{\tilde{\mathbf{b}}}f(\tilde{\mathbf{b}})\\ \tilde{\mathbf{b}}_{t+1}=\arg\min_{\tilde{\mathbf{b}}}\frac{1}{2}\lVert\tilde{\mathbf{b}}-\tilde{\mathbf{b}}_{t+\frac{1}{2}}\rVert_2^2+\eta_{t+\frac{1}{2}} \Omega(\tilde{\mathbf{b}}) \end{cases} \end{align*} They find the minimum of $f$ directly and use the proximal operator of the L1 norm to find a closed form for $\tilde{\mathbf{b}}_{t+1}$. However, $t$ has nothing to do with iterations in this context and I don't see where gradient descent comes in (they say that the formulation was solved using Proximal Gradient Descent). Is it the case that the proximal projection of the minimiser of $f$ is the minimiser of $f+\lambda\Omega$?
Furthermore, the optimisation problem above was already a linear approximation to the original one (Eq 6 in the paper). Are the updates they provide actually the solution to the TCO optimisation problem as claimed?
What am I missing?
Any help is appreciated!