Prediction model with constraint / penalty I am attempting to predict rental price (Sq/ft) for a Retail space. I have a vector of demand / economic variables and other control variables such location and time fixed effects. I'd like to add a constraint to the model for balancing loss due to vacancy.
The data is at monthly level.
Model specification:
Pt = b0 + b1*Dt + b2*Lt + b3*Vt + b4*St + b5*Sup + e 

where, 

Pt = Rental Price
St = Type of the space - categorical variable 
Dt = Demand variables
Lt = Location variables
Sup = Available comparable supply / Vacancy
Vt = Loss due to vacant days ($) 

Vt is the constrain in the model. 
 

I am looking for advise on:

*

*Appropriate model to use (Linear or Non-Linear) and metrics to use for evaluation.

*How to treat constraint variable? Regularization or other approaches?

*Fit a model separately for space type or if included as a variable, how do I estimate the effect of each space type (categorical variable) separately?


I could be misunderstanding the question but I think it's worth distinguishing between whether we are regressing or optimizing. From your equation is looks like $V_t$ is an input variable which you are using to make predictions.
However from your phrasing of the question it sounds more like $V_t$ is itself an function of price and you are looking to maximise price while also trying to minimise the vacant days. This is a bit of a different problem.
If you're trying to predict what the rental price of a space will be given some data then you can postulate the linear relation you defined above and do simple linear regression to estimate the best coefficients.
On the other hand for the second case, you might want to do an optimization to find what price you should set for a particular retail space. This would involve looking for the price which will maximise your profit. Where profit will have some relation like:
\begin{equation}
\textit{Profit}_t=N_{occ}P_t-N_{unocc}C_t
\end{equation}
Where $N_{occ}$ is the number of occupied days, $N_{unocc}$ is the number of unoccupied and $C_t$ is the cost of an unoccupied day. Here we would expect that there on constraints that the total number of days ($N_{occ}$+$N_{unocc}$) is fixed and that their split is in turn affected by the price $P_t$ (more expensive space will have fewer occupied days). We would probably also expect that the number of vacant days for a given price will depend on your other variables (quality of location etc.).
Happy to discuss more in comments if I've misunderstood the question.
