The question is related to strange result from fused lasso estimator
Let us consider fussed lasso estimator:
$$ \hat{\beta}^{FL} = \underset{\beta \in \mathbb{R}^{n}}{\arg \min} [(y_{i} - \beta_{i})^{2} + \lambda_{1} \sum_{i=1}^{n-1}|\beta_{i} - \beta_{i+1}| + \lambda_{2}\sum_{i=1}^{n}|\beta_{i}|], $$
From PATHWISE COORDINATE OPTIMIZATION it follows that the solution is given by the following: let $$ \hat{\beta}^{F} = \underset{\beta \in \mathbb{R}^{n}}{\arg \min} [(y_{i} - \beta_{i})^{2} + \lambda_{1} \sum_{i=1}^{n-1}|\beta_{i} - \beta_{i+1}|], $$ then $$ \hat{\beta}^{FL} = \underset{\alpha \in \mathbb{R}^{n}}{\arg \min} [(\hat{\beta}^{F}_{i} - \alpha_{i})^{2} + \lambda_{2}\sum_{i=1}^{n}|\alpha_{i}|], $$ i.e. first, one can perform fusion, and then we apply lasso to the fussed signal.
The question: is the following procedure correct?
Let us, first, solve the dual to Lagrange's form of fusion for some $s_{1}>0$:
$$ \hat{\beta}^{*F} = \underset{\beta \in \mathbb{R}^{n}}{\arg \min} [(y_{i} - \beta_{i})^{2}] \, \text{ subject to } \, \sum_{i=1}^{n-1}|\beta_{i} - \beta_{i+1}| \leq s_{1}, $$ then, apply lasso to $\hat{\beta}^{*F}$: $$ \hat{\xi} = \underset{\alpha \in \mathbb{R}^{n}}{\arg \min} [(\hat{\beta}^{*F}_{i} - \alpha_{i})^{2} + \lambda_{2}\sum_{i=1}^{n}|\alpha_{i}|]. $$
Then, we can say that $$ \hat{\xi} = \underset{\beta \in \mathbb{R}^{n}}{\arg \min} [(y_{i} - \beta_{i})^{2} + \lambda^{*}_{1}(y,s_{1}) \sum_{i=1}^{n-1}|\beta_{i} - \beta_{i+1}| + \lambda_{2}\sum_{i=1}^{n}|\beta_{i}|], $$
i.e. there exists some $\lambda^{*}_{1}$, which depends on the signal and $s_{1}$.
