Assume $A\in \mathbb{R}^{t\times n}$ and $b\in \mathbb{R}^t$. How to solve the following optimization problem in $x\in \mathbb{R}^n$?
$$\begin{array}{ll} \text{minimize} & \|Ax-b\|_2^2\\ \text{subject to} & x^Tx=1\end{array}.$$
Thanks for the comments, I already read another question. But in fact, the two answers are not complete. In fact, this is not a convex optimization, so the solution we get may be local maximum or something else.
Can anyone prove that if $x,\lambda$ satisfy $$A^TAx-A^Tb+\lambda x = 0\\ x^tx=1.$$ This must be a global minimum?
