# Optimizing Independent Variables to Maximize Dependent Variable

I looked around online and couldn't find anything that was answering my question so I thought I would take to the stack!

I'm interested in knowing if there is a statistical or mathematical way of optimizing ones dependent variable in regression analysis by shifting the values of your independent variables. For instance in market mix modeling generally the regression equation looks something like:

$$\text{Sales = TV Spend + Print Spend + Online Spend + Radio Spend}$$

So once you have your model based on the data how do you optimize the spend levels to have the max sales without just randomly shifting spend from one category to another?

Regression models are not created to do what you want. For example, if you had each of those 4 variables in the model like this: $$y=b0+b1*x1+b2*x2+b3*x3+b4*x4$$ then we can increase the predicted y to infinity by increasing any x variable that has a positive coefficient. Furthermore, we cannot fit the model in one range of observed x values and then use x values way out of that range to make predictions. The indep variables cannot be viewed in isolation -- if you advertise on radio, the impact of additional TV ads will have diminished impact so you need an interaction model. And something to include a decreasing marginal impact of ads in any one area. e.g, A(1-exp(-c*x)) If we created this highly non=linear model that actually described what was going on, then you have a math optimization model (nonlinear programming) to solve: maximize profit from advertizing (must take into account the cost of ads) subject to money < some number and maybe restrictions on min or max money on each medium.