Assume a linear regression problem where I want to force sparsity of some parameters. However, due to some physics, I know that one of my parameters is always positive. For instance, I have that
$$ y=\sum \beta_ix_i+\epsilon $$ where $\beta_5\geq0$
Is it safe to find the parameter estimates through maximizing the penalized likelihood below while just adding the constraint $\beta_5\geq0$
$$l_p=l(\boldsymbol\beta)+\lambda \sum |\beta_i|$$
By safe I mean, can we still interpret the sparsity results the same way we do in the lasso and if yes why, is there another way to do it using an $l_1$ norm, or does this minimization retain the lasso properties at the MLE.