# Optimization based on regression result

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?