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?

  • $\begingroup$ Not sure how to solve this problem, but you need $A$ to have at least one positive entry, otherwise the minimum does not exist. $\endgroup$
    – user1551
    Mar 12, 2019 at 13:32
  • $\begingroup$ Related $\endgroup$ Jun 30 at 10:13

1 Answer 1


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


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