I am stuck with the following problem. Could someone give me some hints? Thanks a lot.
Problem setting: Let $X\in R^{N\times k}$ and $\theta\in R^k$. Now we consider $\hat{\theta} = \arg\min \|X\theta\|^2_2$ with constraint of $\|\theta\|_2=1$. Please solve $\hat{\theta}$.
I have considered the problem as $\hat{\theta} = \arg\min \theta'X'X\theta$ with constraint of $\theta'I \theta = 1$, where we have $I\in R^{k\times k}$.
Now by taking derivative of $\theta'X'X\theta$, we have $2 X'X\theta = 0$ and $\theta'I \theta = 1$. It seems to be a contradiction. Is there any hints for solving the problem? Thanks in advance.