# optimization problem with two lagrange multipliers (proof that OLS is BLUE)

I have some difficulty in understanding the intermediate step to get $$a_i$$.

Attempt: we have the following objective function: $$\sum_{i=1}^n a_i^2 -\lambda \sum_{i=1}^n a_i - \mu\sum_{i=1}^n a_ix_i,$$ where $$\lambda$$ and $$\mu$$ are lagrange multipliers. Differentiating with respect to $$a_i, \lambda$$, and $$\mu$$ yield that $$2a_i -\lambda -\mu x_i$$ and $$\sum_{i=1}^n a_i$$ and $$\sum_{i=1}^n a_ix_i$$, respectively. If we let these equations equal to zero, the second and third equations are just two constraints. I think I should use the first equation to find $$a_i$$, but I am stuck here. Can anyone give me some help?

You have $$\label{eqn1} 2a_i=\lambda+\mu x_i\tag{1}$$ Imposing the constraints, $$n\lambda+\mu \sum x_i = 0,\quad\lambda\sum x_i+\mu\sum x_i^2=2 \sum a_ix_i=2$$ we get $$\lambda=\frac{2\sum x_i}{(\sum x_i)^2-n\sum x_i^2},\quad \mu=\frac{2n}{n\sum x_i^2-(\sum x_i)^2}.$$ Substituting back into \eqref{eqn1} $$a_i=\frac{-\sum x_i+n x_i}{n\sum x_i^2-(\sum x_i)^2}=\frac{x_i-\bar{x}}{\sum (x_i-\bar{x})^2}.$$
• Sorry, but could you tell me how to get $\lambda$? Jun 13 '19 at 11:52
• Solve the simultaneous equations $n\lambda+\mu\sum x_i=0$ and $\lambda\sum x_i+\mu\sum x_i^2=2$. Jun 13 '19 at 11:53
• I multiplied the first equation by $\sum x_i$ and subtract the first equation from the second equation. But, I don't know how to eliminate $\mu$. Could you explain a bit more? Jun 13 '19 at 12:09