Minimal "dominating" rank-$1$ matrix

Question

Given a matrix ${\bf A} \in \Bbb R^{n \times n}$, I would like to find a minimal rank-$1$ matrix ${\bf B} \in \Bbb R^{n \times n}$ such that the Frobenius norm of $ \| {\bf A} - {\bf B} \|_{\text{F}} $ is minimal under the constraint of $ a_{ij} \leq b_{ij}$ for all $i, j \in \{ 1, \dots, n \}$. Also, is there a name for the element-wise "domination" constraint?

Answer 1

$$ \begin{array}{ll} \underset{{\bf x}, {\bf y} \in \Bbb R^n} {\text{minimize}} & \left\| {\bf x} {\bf y}^\top - {\bf A} \right\|_{\text{F}}^2 \\ \text{subject to} & {\bf x} {\bf y}^\top \geq {\bf A} \end{array} $$