Hy everybody,
I am trying to optimize a calibration process I am developing and I want to filter the data to minimize the error in my calibration. I have been able to simplify the problem all the way to the following constrained optimization problem:
Given two sequences $\{a_i\}$ and $\{b_i\}$ which are:
- finite
- positive
- real
find the sequence $\{x_i\}$ which respects the constraint $\sqrt{\sum_ix_i^2}=1$ and minimizes:
$$\frac{\sqrt{\sum_i a_i^2 b_i^2 x_i^4}}{\sum_i a_i^2 x_i^2}$$
I attempted to solve this problem using the method of Lagrange multipliers but I got stuck in trying to solve the system of equations.
(Note also the constraint $\sqrt{\sum_i a_i^2x_i^2}=1$ is fine for my application and it seems to make the calculations a bit easier)
Any help is appreciated, Thanks!
Hint.
Assuming $\{a_k\ne 0,\ b_k\ne 0\}$ for $\{k = 1\cdots n\}$ calling $Y = (y_k),\ Z = (z_k),\ \{k = 1\cdots n\}$ and making $a_k^2x_k^2=y_k,\ b_k^2x_k^2= z_k$ we have the lagrangian
$$ L(Y,Z,\lambda) = Y\cdot Z+\lambda_0(Y\cdot \mathcal{1}-1)+\sum_{k=1}^n\lambda_k\left(\frac{y_k}{a_k^2}-\frac{z_k}{b_k^2}\right) $$
where $\mathcal{1}=(1,\cdots,1)$. Here the system of equations to determine the stationary points, is linear.
$$ \nabla L = 0 = \cases{z_k+\lambda_0+\frac{\lambda_k}{a_k^2}\\ y_k-\frac{\lambda_k}{b_k^2}\\ \frac{y_k}{a_k^2}-\frac{z_k}{b_k^2}\\ \sum_{j=1}^n y_j - 1 } $$ with $3n+1$ equations and $3n+1$ unknowns.
NOTE
$$ \cases{ \lambda_0 = -\frac{2}{\sum_k\frac{a_k^2}{b_k^2}}\\ \lambda_k = -\frac{\lambda_0}{2}a_k^2 = \frac{a_k^2}{\sum_k\frac{a_k^2}{b_k^2}}\\ y_k = \frac{a_k^2}{b_k^2}\frac{1}{\sum_k\frac{a_k^2}{b_k^2}} = a_k^2x_k^2 } $$
etc.