General Markowitz-style optimization (problem objective of $w^T \mu - \lambda w^T \Sigma w$) yields simple optimal weights policy $w \propto \Sigma^{-1} \mu$.
However, I would like to add a series of factor exposure constraints (i.e. $| w^T \beta_i | < k_i$ for factors indexed by $i$, instrument exposure vector to $i^{th}$ factor by $\beta_i$). I may wish to extend to maximum position values as well, which is basically just another constraint with a different beta vector.
Is there a standard way do this quickly, closed form or similar, with the addition of these types of constraints? Or some common, clever transformations/relaxations that would help facilitate practical solutions under time constraints?
I've got to imagine this is a pretty solved problem, since many portfolios have these types of constraints. I'm just not sure if folks throw into an optimizer and don't care about very fast solutions/closed form solution explicitly. Or do something altogether different. Is there some industry standard or set of resources available to follow? Thanks!