This is a general question about how linear programming is used in the analytics community.
Is it common, or feasible to use linear regression (or perhaps even more complex models like regression trees) to act as the objective function in a linear program?
I'm interested in statistically deriving linear models of complex phenomena (i.e energy transfer through a surface) and finding optimal combinations of the variables using some optimization methods. Here's an example to clarify my question.
** UPDATED (SO THERE'S NO TRIVIAL SOLUTION) **
Let's say I fit a regression to some heat transfer data, and derive a regression line that can predict the mechanical heating of a room (Qh).
So my regression might look like this:
Qh = b0 + (b1 * km) + (b2 * kg) + (b3 * Am) + (b4 * Ag) + (b5 * dT) + (b6 * Qr) Where: Qh = Mechanical heating of the room (Wh) Qr = Solar radiation flux transmitted through glass (W/m2) km = Conductance of masonry (low) (W/m2/K) kg = Conductance of glass (high) (W/m2/K) Am = Area of masonry (m2) Ag = Area of glass (m2) dT = Temperature difference between outside and inside (K) b0, b1, b2, b3, b4, b5, b6 = regression coefficients.
Solving the regression might give us something like this:
Qh = 1 + (2 * km) + (3 * kg) + (4 * Am) + (5 * Ag) + (6 * dT) + (7 * Qr)
(Note that I'm deliberately not going to use variables to represent the regression weights since they're not variables in this case, the regression has been solved so they are constants).
I would like to find the optimal combination of the wall/glass area that reduces mechanical heating of the room. So the variables for the linear program are (Am, Ag) and we assume everything else is a constant.
This problem is somewhat tricky since, reducing the (high conductance) glass area will reduce heat loss through the wall, and reduce mechanical heating - but will also reduce the transmitted solar radiation that also will reduce mechanical heating.
Can I therefore create a linear program that finds this for me?
In my linear program, this new optimization problem would be represented as:
Objective_function = min(Qh = 1 + (2 * km) + (3 * kg) + (4 * Am) + (5 * Ag) + (6 * dT) + (7 * Qr)) Variables = Am, Ah (everything else would be a constant determined by the user). Constraints: 0 < Am < 10; 0 < Ag < 10; Am + Ag = 10.0
** END UPDATE **
Could I use my regression as a simulation (aka surrogate model) and find the optimal combination of a the variables this way via linear programming? My sense is that a linear program would be uniquely suited for this kind of problem, since it can only represent linear relationships.
However, after some google/stack overflow searches, I haven't been able to find any examples of this particular combination. I'm getting a lot of hits about using linear programming to optimize the regression itself (i.e to minimizing the cost), but not about it's use as the objective function.
Is this just because the use of regressions in linear programming is so obvious, and self-evident no one needs to mention it explicitly? Or am I missing something about why regressions aren't used in linear programming?