When simplified down, my optimization problem has the following structure
$$\underset{x\in\mathbb{R}^n}{\arg\min} \quad \left\| b - A\operatorname{vec} \left( x x^\top \right) \right\|_2^2$$
I am interested in the following equivalent (?) problem
$$\underset{Y\in\mathbb{R}^{n\times n},\ Y\succeq 0,\ \operatorname{rank}(Y)=1}{\arg\min} \quad \left\| b -A \operatorname{vec}(Y) \right\|_2^2$$
Introducing the positive semidefinite matrix $Y$ with rank 1 constraint makes the cost function quadratic, which is convenient for later analysis. However, I struggle with the question of all points that satisfy first-order optimality conditions are the same for both problems. In other words, if $Y^\star$ satisfies first-order optimality conditions, does $\tilde{x}$ with $Y^\star = \tilde{x}\tilde{x}^\top$ also satisfy first-order optimality conditions for its respective optimization problem.