Parameter optimization using a regression model.

Question

I am working on an optimization problem. I build a regression model to understand the behavior of a system which depends on two variables which are functions of another two variables. My regression model is as follows

$Time$ =$\alpha_1$.$M$ + $\alpha_2$.$L$ + $\eta$

Now I know that $M$=$f(N,BW)$ and $L$=$f(N,Const)$

$Const$ is something we can't change and always exist within the system. My goal is to find $N$ and $BW$ in order to minimize $Time$ for a given $\alpha_1$ and $\alpha_2$. Under the following Constraints:

1.$N$ can be a positive whole numbers where $N<=N_{max}$.

2.$BW$ can be any positive number where $BW<=BW_{max}$.

I computed $\alpha_1$ and $\alpha_2$ for an arbitrary $N$ and $BW$ but don't have idea how to find the optimal values for $N$ and $BW$.

Any help is much more appreciated. Thanks!

Answer 1

It could look like

$$\underset{N,BW}{\texttt{min}} \ \ \sum_{i=1}^n \left[ T_i- \alpha_1(N\cdot n_{i}+ BW\cdot b_{i})-\alpha_2(N\cdot n_{i}+C )-\eta\right]^2 $$

s.t.

$N\leq N_{max}$

$BW\leq BW_{max}$

$N,BW \in \mathbb N$

n is the number of observations.

$T_i$, $b_i$ and $n_i$ are the observations.

The parameters $N$ and $BW$have to be determined.

Answer 2

So let's assume the linear case:

$M=\gamma_1N + \gamma_2BW$

$L=\beta_1N + C$

Now plug these into your regression equation: $Time = \alpha_0 + \alpha_1M + \alpha_2L + \eta$

which yields:

$Time =(\alpha_0+\alpha_2C) + N(\alpha_1\gamma_1 + \alpha_2\beta_1) + \alpha_1\gamma_2BW + \eta$

In the linear case, it is easy to see that the minimum is a function of the signs on the coefficients on $N$ and $BW$. E.g. if $(\alpha_1\gamma_1 + \alpha_2\beta_1)$>0 we want N as small as possible and vice versa.