Optimization based on regression result

Question

I am trying to find the optimal values for a given attributes. In particular, my objective is to maximize the profitability based on some parameters. If we call the profitability $p$, and the parameters $x_1,x_2,...,x_n$, I run a regression model and obtain the relationship between the response ($p$), and the parameters $x_1,x_2,...,x_n$: $p = \beta_0 + \beta_1 x_1 + ...+\beta_n x_n$. To capture the relation between the X's, I regress each $x_i$ on $x_j, j = 1,2,...,n, j\neq i$, to obtain $x_1 = c_0 +c_1 x_2+ c_2 x_3+...+c_{n-1} x_n$, $x_2 = d_0 +d_1 x_1+ d_2 x_3+...+d_{n-1} x_n$, $x_n = e_0 +e_1 x_1+ e_2 x_2+...+c_{n-1} x_{n-1}$.

Then I set up my optimization problem as follows:

$max \hspace{0.25 cm} (\beta_0 + \beta_1 x_1 + ...+\beta_n x_n) \\ s.t. \hspace{0.5 cm}x_1 \geq \epsilon+c_0 +c_1 x_2+ c_2 x_3+...+c_{n-1} x_n \\ \hspace{1.288 cm}x_2 \geq \epsilon+d_0 +d_1 d_1+ d_2 x_3+...+d_{n-1} x_n\\ \hspace{5 cm}.\\ \hspace{5 cm}.\\ \hspace{5 cm}.\\ \hspace{1.288 cm}x_n \geq \epsilon+ e_0 +e_1 x_1+ e_2 x_2+...+c_{n-1} x_{n-1}\\ \hspace{1.288 cm}x_1 \leq \epsilon -c_0 +c_1 x_2+ c_2 x_3+...+c_{n-1} x_n \\ \hspace{1.288 cm}x_2 \leq \epsilon-d_0 +d_1 d_1+ d_2 x_3+...+d_{n-1} x_n\\ \hspace{5 cm}.\\ \hspace{5 cm}.\\ \hspace{5 cm}.\\ \hspace{1.288 cm}x_n \leq \epsilon- e_0 +e_1 x_1+ e_2 x_2+...+e_{n-1} x_{n-1}$

I tried the above method, but I keep getting that the solution is infeasible. Now my questio is: Is there any problem with my method? any alternative ideas?

Answer 1

Pardon me, but what the heck are you doing?

If you want to optimize the objective, you should determine the actual constraints which constrain the parameters whose optimal values you are trying to find. If there is a cost component, then either include that as part of the objective function (for example, revenue minus cost), or maximize revenue subject to some cost (and possibly other) constraint.

The constraints as you have provided them do not seem to make any sense in the context of an optimization problem. It looks like you're trying to convert approximate regression relations between the parameters, and then trying to impose some constraints based around those, perhaps with some wiggle room, in a way which doesn't seem to make any sense. So follow the advice in the preceding paragraphs.