I am currently reading a paper stating the following regression problem $$\text{min} \sum_{i=1}^N ||\beta\cdot x_i-y_i||\\ \text{subject to} \sum_{j=2}^M ||\beta_{j}-\beta_{j-1}|| \leq S $$ for vectors $x_1,\dots,x_n, \beta \in \mathbb{R}^n, y_i \in \mathbb{R}$. This is a slightly changed version of the fused lasso loss for a regression problem. The authors state, this poblem can be solved via maximum likelihood estimation. I would have used quadratic program solvers instead but they are slower of course. Are there closed form solutions (from MLE) for this kind of problem?

I am currently reading a paper stating the following regression problem $$\text{min} \sum_{i=1}^N ||\beta\cdot x_i-y_i||\\ \text{subject to} \sum_{j=2}^M ||\beta_{j}-\beta_{j-1}|| \leq S $$ for vectors $x_1,\dots,x_N, \beta \in \mathbb{R}^n, y_i \in \mathbb{R}$. This is a slightly changed version of the fused lasso loss for a regression problem. The authors state, this poblem can be solved via maximum likelihood estimation. I would have used quadratic program solvers instead but they are slower of course. Are there closed form solutions (from MLE) for this kind of problem?